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G of the natural projection G-> G/H, (see Appendix A), so [cp(x) ] = x for all x E G/H Define A = { (s, x)E S x G/H i sEB cp(x) } Then A is Borel, and since cp(xg) = hcp(x)g for some h E H, A is clearly G.-invariant If B is neither null nor conull, the same is true for A. Corollary [Moore 1 ] : If H 1 , H 2 c G are two closed subgroups, then H 1 is ergodic on G/H 2 if and only if H 2 is ergodic on G/H 1 .
2.2.3
Proof: By the proposition, both assertions are equivalent to the ergodicity of
G on G/H 1
x
G/H 2 .
Moore's ergodicity theorem is a very general result which completely answers Problem 2.2J(a) for G a simple Lie group (such as SL(n, IR)) where H 1 or H2 is a lattice in G. It also provides an equally complete answer if G is a suitable semisimple Lie group and r c G is an irreducible lattice . Definition: Suppose G is a connected semisimple Lie group with finite center and r c G is a lattice . Then r is called irreducible if for every non-central normal subgroup (equivalently, every normal subgroup of positive dimension) N, r is dense when projected onto GjN. 2.2.4
This definition is of course designed to eliminate examples such as X r2 c G 1 X G 2 A typical example of an irreducible lattice is the following
r1
2.2.5 Example: Let G = SL(n, IR) x SL(n, IR), and(!) = Z [j2]. For a+ bj2E(!), let Y is Bore[ and essentially G-invariant, then f is essentially constant.
2.2.18
2..216, and the fact that for any countably there is an injective Bore! map Y ----> [0, 1] (Appendix A)
Proof: This follows from 2.1.1 1 ,
separated
Y
By Corollary 2..2. 1 7, Moore's theorem can be converted to a problem about unitary representations. Namely, if n is a unitary representation of G with no invariant vectors, and H c G is a closed subgroup, we want to know when H can have invariant vectors. To prove Theorem 2..21 5 it clearly suffices to prove the following Theorem [Moore 1 ] : Let G; be a connected non-compact simple Lie group with finite center, G = ITG; (finite product), and suppose n is a unitary representa tion of G so that for each G;, n l G; has no invariant vectors. If H c G is a closed subgroup and n I H has non-trivial invariant vectors, then H is compact.
2.2.1 9
This result in turn follows from the following more general theorem. If n is a
23
Moore's ergodicity theorem
unitary representation of G on a Hilbert space £, and v, wE £ are unit vectors let f(g) = 0 as g leaves compact subsets of G
2.2.20
Theorem 2.2. 1 9 follows from 2..2. 20 because if n ! H has an invariant vector v, the matrix coefficient 0, then n I N has invariant vectors (namely :Yfo). We now show that if f.l( {0} ) = 0, then assertion (ii) in the theorem is satisfied . To see this, consider
the action of P on N.. An elementary calculation shows that
Ergodic theory and semisimple groups
28
acts on N
�
lP via multiplication by a 2 . Hence, given any compact subsets
F c lP - {0}, for g E A outside a sufficiently large compact set we have Jl(gE n F) = 0 . Given any two unit vectors f, h E L 2 (1P, J1, {£'A } ), and 8 > 0 we can choose compact subsets E, F c lP - {0} such that E,
.
11 XE f - f 1 1 � 8 and 11 X F h - h 11 � 8 Then
l W is a regular
37
Algebraic groups and measure theory
map. Suppose there is a set A Then f is defined over k
c
Vk which is Zariski dense in V such that / (A) c Wk.
Proof: It suffices to consider the case in which both V, W are affine, and by considering coordinate functions, the case in which W = K Then we can write f = fo
r
+ L
i= 1
ex;
f; where /o, [; are polynomials with coefficients in k and
ex; E K
with { 1 , ex 1 , . . , ex, } linearly independent over k . Since j (A) c k, it follows from the linear independence of { 1 , ex 1 , . . . , ex, } that f (a) = fo (a) for all a E A Since A is Zariski dense in V, f = f0 on V, verifying the proposition.
If/ :G __. G' is a regular map of algebraic groups which is an isomorphism of abstract groups, then f l is also regular, i.e . , f is an isomorphism of algebraic groups. From 3J JO we obtain: -
3.1.1 1 Jff is
Corollary: Suppose G, G' are k-groups and f :G __. G' is an isomorphism. defined over k, so is f -l
We recall that a connected algebraic group G is called semisimple if the radical of G (i . e. the maximal normal connected solvable subgroup) is trivial. If G is a k-group, G is called k-simple if every proper normal k-subgroup is trivial and almost k-simple if every proper normal k-subgroup is finite . (If G is a k-group, so is its radical, and hence semisimplicity does not depend upon a field of definition k c K ) A regular (k-) homomorphism f :G __. H is called a (k-) isogeny if it is surjective with finite kerneL For every connected semisimple G there exist algebraic groups G, G' and isogenies G� G �----?7 G' with the following properties. For every isogeny H __. G, there is an isogeny G ...... H such that the composition G __. H ...... G is n:1 . For every isogeny G ...... H, there is an isogeny H __. G' such that the composition G __. H ...... G' is n: 2 . G is called the (algebraic) universal covering of G and G' the adjoint of G . If G is a k-group, so are G and G' and n:1 and n2 are defined over k. G is called (algebraically) simply connected if G = G and is called an adjoint group if G = G' For example, SL(n, C) is (algebraically) simply connected and PSL(n, C) is an adjoint group. If G is a connected semisimple k-group which is either simply connected or adjoint, then G � ITG; as k-groups where G; is an almost k-simple k-group. If G is adjoint, then G; are k-simple. (See [Tits 2] for further details. ) Finally, suppose k c K are fields of characteristic 0, K algebraically closed Let G c GL(n, K) be a k-group. Suppose now that we also have k c n where Q is algebraically closed. Then taking the Zariski closure of Gk in GL(n, Q) we
38
Ergodic theory and semisimple groups
obtain a k-subgroup Gn of GL(n, Q). Since the connected component of the identity and the radical of G are both defined over k, Gn will be connected or semisimple if and only if G has the same property . For example, suppose G c GL(n, IC) is a connected semisimple (Q-group If (Q P is the field of p-adic numbers with algebraic closure (lP, then G10, will be a connected semisimple (Q group, and in particular, a connected semisimple (Qp-group This construction will arise in Chapter 6 In this context, the following result is useful. 3.1.12 Theorem: Let G be a semisimple (Q-group. Then for all but finitely many primes p, G10, is not compact.
For a proof, see [Springer 3, lemma 4 9J 3.2
Orbits of measures on projective varieties and the Borel density theorem
Suppose X is a compact metric space . Let M(X ) be the space of probability measures on X, and C(X ) the continuous functions on X Then we can identify M(X ) as a closed subset of the unit ball in the dual space C(X ) * , where the latter is given the weak- *-topology.. Thus, M(X ) is a compact metrizable space with this topology.. If J.ln, J.LE M(X ), we have J.ln -+ J1 if and only if JfdJ.tn -+ J fdJ.t for all /E C(X }. If G acts continuously on X (i.e . , the action map X x G -+ X is continuous), then G will act continuously on M(X ), via the action (J.t g)(A) = J.t(Ag - 1 ) for A c X Bore!, or alternatively, via the equation Jfd(J.t g) = J (g f )dJ.t, for /E C(X ), where (g f )(x) = f ( xg) We shall be particularly interested in this action when G is a semisimple Lie group and X = G/H for some algebraic subgroup H, or more generally for groups of the form Gk ( G algebraic over k) where X = Gk/Hk for some k-subgroup H c G. The space M(X ) will of course in general be infinite dimensional. Nevertheless, the main results of this section will show that in certain respects, the action of Gk on M(Gk/Hk) behaves much like the action of Gk on a v ariety Namely, the stabilizers are close to being algebraic and in fact are algebraic for k = IR, (first shown when H is minimal parabolic in [Moore 2] ), and the orbits are locally closed [Zimmer 4]. For G = PGL(n, k) acting on M(IP" - 1 ), an examination of the stabilizers will lead to an easy proof, due to [Furstenberg 4], of the Bore! density theorem [Bore! 3]. Let k be a local (i.e., locally compact, non-discrete) field of characteristic 0. If V c k" is a non-zero linear subs pace, we denote by [ V] c lP" - 1 (k) its image in IP" - 1 We let PGL(n, k), as usual, be the pr�jective general linear group, ie , GL(n, k)/k * I, where k * is the set of non-zero elements of k.. If g E GL(n, k), we
39
Algebraic groups and measure theory
denote by [g] its image in PGL(n, k). There is a natural action of PGL(n, k) on IP" - 1 (k), and hence on M(IP" - 1(k)). The following will be very useful
[Furstenberg 1]: Suppose [gm ] E PGL(n, k), f.l, vEM(IP" - 1 (k)) and that f.1 [gm ] ----> v . Then either (i) { [g m ] } is bounded, i e , has compact closure in PGL(n, k); or (ii) there exist linear subspaces V, W c k", 1 � dim V, dim W � n - 1, such that v is supported on [ V] u [ W] 3.2.1
Lemma
k, and then a norm on the k-linear space M(n x n, k) of n x n matrices over k. Since we are free to multiply gm by a non-0 scalar, we can assume that 11 gm 1 1 is bounded and bounded away from 0. Let us assume that { [g m] } is not bounded. Then by passing to a subsequence, we can assume [gm ] ----> oo in PGL(n, k), i . e. , eventually leaves every compact set in PGL(n, k) . Since 11 gm 1 1 is bounded, again by passing to a subsequence we can assume gm ----> g E M(n x n, k). Since [gm] ----> CXJ, g� GL(n, k). On the other hand, since 11 gm 1 1 is bounded away from 0, g i= 0. Thus, if we let N = kernel(g) and R = image(g), then 1 � dim N, dim R � n - 1 . Since the Grassmann varieties are compact, again by passing to a subsequence, we can assume [N ] gm ----> [V] for some V, dim V = dim N. We also observe that if x E lP" - 1 , x � [N] then lim x gm = x g E [R]. We can write f.1 = f.1 1 + f.lz where f.1 1 (1P" - 1 - [N ] ) = 0 and f.lz( [N ] ) = 0. We can assume by compactness of M(IP" - 1 ) that f.ligm ----> v;, i = 1 , 2, and we have v = v 1 + v 2 Since f.1 1 is supported on [N ] and [N ] gm ----> [ V], it follows that v 1 is supported on [ V]. Thus to prove the lemma, it suffices to show that v 2 is supported on [R]. Suppose /E C(IP" - 1 ) and that/ = 0 on [R]. Then Proof: We can choose an absolute value on
J fd vz = lim J fd(f.lz g m ) = lim J ' ·
" r '
f(xgm)df.lz(x)
.
However f.lz[N] = 0, so this is Sur ' - iNd(xgm)df.lz(x).. But since f = 0 on [R], f(xgm ) ----> 0 pointwise on IP" - 1 - [N ], and hence the integral converges to 0 by the dominated convergence theorem. This completes the proof of the lemma.
Corollary: For f.l E M(IP" - 1 ), let PGL(n, k)p. be the stabilizer of f.1 in PGL(n, k). Then either
3.2.2
(i) f.1 is not supported on a finite union of proper projective subspaces, in which case PGL(n, k)" is compact; or (ii) There is a proper subspace V0 c kn such that (a) f.l[ Vo] > 0, and
40
Ergodic theory and semisimple groups
(b) PG L(n, k)JL( [ V0 ] ) is a finite union of projective subspaces [ V0 ] , [ V,]
In particular, PGL(n, k)�' has a subgroup of finite index leaving Vo invariant Proof: (i) is an immediate consequence of Lemma 3 2. 1 To see (ii), let V0 be a subspace with minimal dimension so that Jl( [ V0 ] ) > 0. For g E PGL(n, k)�', Jl(g [ V0] ) > 0. Furthermore, if Jl(g[ V0] n [ V0] ) > 0, it follows by the minimality of dimension property of V0 that g[ V0] = [ V0] . Since J1 is a probability measure, it follows immediately that {g [ Vo] l g E PGL(n, k)JL } must be a finite collection of projective subspaces . Thus, (ii) follows as well.
From this corollary, we can see that the stabilizers of measures are almost algebraic. More precisely, let us make the following definition
3.2.3
Definition: Suppose k is as above, a local field of characteristic 0.. Suppose
G c GL(n, k). We call G k-almost algebraic if there is an algebraic k-group H such that Hk c G is a cocompact normal subgroup .
We recall that compact real matrix groups are IR-points of algebraic IR-groups . (See, e .g., [Baily 1]) Thus, for k = IR, any k-almost algebraic group is actually (the IR-points of) an algebr aic group (over IR). For k = IR, a result of the following type was first observed in [Moore 2]. Theorem: For any measure Jl E M(IPn - 1 (k)),
the stabilizer PGL(n, k)JL has a normal subgroup of finite index which is k-almost algebraic. In particular, for k = IR, PGL(n, IR)p is itself the real points of an IR-group . 3.2.4
Proof: As it should cause no confusion, for this proof we shall refer to the
k-points of a k-group simply as an algebraic group. For any J1E M( IPn - 1 (k)) we can find a countable family of (positive) measures { u;} on IP n - 1 (k) of total measure at most 1 such that: (a) for i # j, Jl; l_ Jli; (bl LJli i
= jl;
(c) if V; is of minimal dimension among all linear subspaces with Jl;( [ V] ) > 0, then supp(Jl;) c [ V;];
41
Algebraic groups and measure theory
(d) with [ V;] a s i n (c), if i # j, then [ V;] # [ ViJ As in 3. 2.2, each f.1; must have a finite orbit under PGL(n, k)� Replacing each by Lf.li where f.liE PGL(n, k)�f.l;, and eliminating repetitions, we obtain a countable family of measures { v; } such that: f.1;
( 1 ) v; l_ vi for i # j;
(2) L V; = f.1; (3) PGL(n, k)� = n PGL(n, k) v,; i
(4) if V; is of minimal dimension among all linear subspaces for which v;( [ V;] ) > 0, then PG L(n, k)i [ V;] ) is a finite union { [ W;i],j = 1, . . , n;} of projective subs paces of IP"- 1 (k), and v ; is supported on PGL(n, k)i [ V;] ). .
For each i, let H; = {gE PGL(n, k) l g[ W;i] c PGL(n, k)i [ V;] ) for all j} Let N; = { g E PGL(n, k) i g l W;i is scalar for all j} Then H;, N; are algebraic, N; c H; is a normal subgroup, N; c PGL(n, k)v, and PGL(n, k)� c H;. Thus
n N; c PGL(n, k)� c n H;. Since N;, H; are algebraic, by the descending chain i
condition on algebraic groups we can find a finite set of i, say F, such that
nN; = n N; and nH; = nH; For i E F, let H i = {gEPGL(n, k) i g[ W;iJ = i
iEF
i
ieF
[ W;i] for all j}. Then N; c H i c H;, and H i is a normal algebraic subgroup of finite index in H;. Letting
G
= PGL(n, k)� n nHi, we have that G is a normal ieF
subgroup of finite index in PGL(n, k)�'. Therefore, it suffices to show that
GfnN; is compact ieF
We have a natural homomorphism n : n H ;;nN; � ieF
ieF
n ieF, j
PGL( W;i) which is
regular and hence has a closed image Since it is also injective, it is a homeomorphism onto its image. Thus, to see that G/n N; is compact, it suffices to see ieF
that the projection of n(G/n N;) into each factor PGL( W;i) is precompact But iEE
this image leaves the measure v ;I [ W;i] on IP(W;i) invariant, and by conditions (3), (4), and Furstenberg's lemma (3 2. 1 ), this image is precompact We now turn to Furstenberg's proof of the Borel density theorem, first proved in [Borel 3] 3.2.5
Theorem
(Bore/ density theorem, [Bore/ 3] ): Let G be a connected semi-
Ergodic theory and semisimple groups
42
simple algebraic P--group, G = G�, and assume G has no compactfactors. Let r c G be a (topologically) closed subgroup such that Gjr has afinite G-invariant measure. Then is Zariski dense in G (ii) r 0 (the connected component in the Hausdorff topology) is normal in G. In particular, if G is simple and r is a proper subgroup, then r is discrete (i)
r
Proof: Let H be the Zariski closure of r in G and H = H n G.. By 3.. L9, it suffices to see that H = G . Since r c H, we have a G-map Gjr --;. G/H, and so G/H also has a finite G-invariant measure. By Chevalley's theorem (3JA), there is a representation G --;. SL(n, P-) for some n, and a point x E IP"- 1 such that H = Gx, i . e. the stabilizer of x in G. By passing to a subspace, we can clearly suppose that the linear span of x· G is P-". If n = 1, then the representation of G is trivial, H = G, and we are done. If n � 2, the image of G in SL(n, P-) does not have compact closure.. (This follows from the well-known fact that G does not have non-trivial finite-dimensional unitary representations. This assertion follows from 2. 2. 20 for example, although simpler proofs are available. ) However, the map G --;. IP" - 1 , g -;. x g, induces a map G/H --;. [p" - 1 , and since G/H has a finite invariant measure, there is a measure flE M(IP" - 1 ) left invariant by G. Choose V0 as in Corollary 3.2. 2. By connectedness of G, G leaves [ Vo] invariant. Since /l is supported on x G and /l( [ Vo] ) > 0, it follows that x · G n [ Vo] # 0, and by invariance, that x G c [ V0]. Since Vo is a proper subspace, this contradicts the fact that x · G spans P-". Thus, (i) is established. (ii) let L(G) be the Lie algebra of G, and L(r0 ) c L(G) the Lie algebra of r0 . Since r o c r is normal, L(r0 ) is Ad(r)-invariant But Ad is a rational representation of G, so by (i), L(r0 ) must also be Ad(G)-invariant Thus, r o is normal.
Remark: The same proof shows the following result due to [Wang 2]. Let k be
a local field of characteristic 0, G a connected semisimple k-group such that for every simple k-group H is which a factor of G, Hk is non-compact Let r c Gk be a closed subgroup (in the Hausdorff topology) such that Gkjr has a finite G-invariant measure.. Then r is Zariski dense in G . Thus f1u, our results in this section have concerned the stabilizers of measures in M(IP" - 1 (k)).. We now turn to consideration of the orbits as subsets of M(IP" - 1 (k)). Our next goal is the following.. 3.2.6
Theorem
[Zimmer 4]: Let k be a local field of characteristic 0 . Then every
43
Algebraic groups and measure theory
PGL(n, k)-orbit in M(IP" - 1 (k) ) is locally closed, and hence the action is smooth We begin the proof by establishing some notation . Let d be a metric on - 1 (k) Let re be the space of closed non-empty subsets of lP" - 1 (k) Since lP" - 1 (k) is compact, re will itself be a compact metric space with the Hausdorff metric lP"
d(A, B) = sup {d(x, B), d(y, A) } . If A E({/ we have a natural identification XEA, yEB
of M(A) with the measures on IP" - 1 (k) which are supported on A With this identification, we shall write M(A) c M(IP" - 1 (k))..
3.2.7
Lemma: If A,
then ,uEM(A).
A;E({/ with A; ---+ A, and Jl; EM(A;) with J1; ---+ ,UE M(IP" - 1 (k)),
The proof of the lemma is a straightforward exercise. If d c re, let M91
3.2.8
Lemma:
=
U M(A). From Lemma 3..2. 7, we have:
AE 91
If d c re is closed, then M x� c M(IP" - 1 (k)) is closed.
We shall now fix a choice of d
c
re Namely, let d = { A E({/ I A
c
U [ V;] i= 1 for some linear subs paces V; c k", such that for all i # j, V; if- Vh and L dim V; ;£ n}. Then i t i s straightforward that d c re i s a closed subset For A E d, define: n(A) = .£, d(A) = L dim V;, and D(A) = dim (L V;) Then we have 1 ;£ n(A), d(A), D(A) ;;;; n. The following two lemmas are again straightforward to verify.
3.2.9
Lemma:
=
Let
de = {A E d i n(A) ;;;; .£ }, dd = { A E d l d(A) ;;;; d}, d(D) = {A E d i D(A) ;;;; D}. Then these are all closed subsets of d . Lemma: Let gut = {A Ed I n(A) = .£, d(A) = d}. If AiE!!Bt, and Ai -+ A E re, then A Eggt u d d - 1 Hence 2B1 u d d 1 is closed.
3.2.10
-
We are now ready to prove the theorem
44
Ergodic theory and semisimple groups
Proof (of Theorem 3.2.6): Let p E M(Pn - l (k)) Define
d(p) = min { d (A) I A E d and pE M(A) } . Let
n(p) = max { n(A) I A E d JlE M(A), and d(A) = d(p) }. Fix A E d with pE M(A), d(A) = d(p) and n(p) = n(A).. Then define D(p) = D(A) f
We observe that if A = U [ ViJ, then J1 is also supported on [I: VJ Thus, by i= 1 the definition of d(p) and the choice of A, we have d(A) = D(A), i e , the subs paces Vt , . , Vr are independent Now let K(p) c 'l/ be defined by
By Lemmas 32. 9, 3.210, K(p) is closed.. Finally, we define Oli (p) = M(Pn - 1 (k)) - M K(f'l·· Then p E Oli (p), Oli (Jl) c M(Pn - 1 (k)) is open by Lemma 3 2.8, and Oli (p) is readily seen to be PGL(n, k)-invariant To show that Orbit(p) is locally closed, it suffices to see that Orbit(p) n Oli (p) = Orbit(p).. By our observations of the preceding sentence, Orbit(p) c Orbit(p) n Oli(p).. To establish the reverse inclusion, let v E Oli (p) and assume there is a sequence [gi] E PGL(n, k) such that [gi] J1 """""* v. We want to show v E Orbit(p) Recall the set A = u [ ViJ defined above . By passing to a subsequence, we can assume that for each j, [gi] [ VJ """""* [ WiJ for some Wi with dim Vi = dim Wi. Thus by Lemma 3..2. 7, v E M(u [ WiJ ). Since v E Oli (p), dim(I: Wi) = D( u [ WiJ ) � D(p) (by definition of K(p)) = d(p) = I: dim vi = I: dim wj In other words, the subspaces Wi are independent as well Let Jli = 11 1 [ Vi] Again by passing to a subsequence, we can assume that for each j, [gil Jli ---* Vj, where v i E M( [ Wi J ). For each i, j, let gi( Vi """""* kn be gii = gi l Vi. We will apply the following variant of Furstenberg's lemma (3.2.1) to the sequence { gii l i E Z } . Lemma: Let M(n x r, k) b e the set of n x r matrices over k, a n d let r, k) be the matrices corresponding to injective linear maps k' """""* kn . Then x r, k) is an open subset of M(n x r, k), and the non-0 scalars k * acts by
3.2.1 1
I(n I(n
x
45
Algebraic groups and measure theory
multiplication on I(n x r, k) For fE I(n x r, k), let [f] : IP' - 1 (k) -. IP" - 1 (k) be the induced map . If J1 E M(IP' - 1 (k) ) and j; E I(n x r, k) with [f;] * J1 -> V E M (IP" - 1 (k)), then either (i) { [f;] } is bounded in I(n x r, k)/k * or (ii) v is supported on [ V] u [ W] where 1 :;:; dim V, dim W :;:; n 1 and dim V + dim W = r. -
(We recall that if f: X -> Y and J1EM(X ), then (f* J1)(B) = J1(/ - 1 (B)).) The proof of Lemma 12. 1 1 is virtually identical to that of Lemma 3.2. 1 We now complete the proof of the theorem. We claim that for each j, { [ O ;i] ) i E Z } is bounded in I(n x dim Vh k)/k * . Suppose this fails for some j Then by Lemma 3. 2. 1 1 , vi is supported on [ YiJ u [Z iJ where dim Yi + dim Zi = dim Wi. If Yi n Zi # 0, then replacing [ Wj] by [ Yi + Zi] in the support of v, we see that v E M s4(D(I'l - 1 l which is impossible by the definition of K(Jl) (recalling that V E 0li(J1)) . On the other hand, if yi n zi = 0, a similar replacement implies that V is supported on an element of .'?B���l + 1 , which is likewise impossible . Thus, the boundedness of the above sequences is verified Therefore, by passing to a subsequence we can assume there are ciiEk * such that for each j, lim c;i% = hi where h( Vi -. k" is an injection. In this case, we i
clearly have [hiJJli = vi and hi(Vi) = Wi. Since { Vi} and { Wi} are both independent collections of subspaces, we can clearly find h E GL(n, k) such that h i Vi = hi· · Then [h] J1 = v, and v E Orbit(Jl) as required. This completes the proof of Theorem 3.2 . 6. More generally, we can consider the Gk-orbits in M(IP" - 1 (k)) where G is an algebraic k-group. We then have the following consequence of Theorem 3..2. 6 Corollary [Zimmer 4]: Suppose k is a local field of characteristic 0 and G c GL(n, K) is a k-group . Then the action of Gk on M(IP" - 1 (k)) is smooth
3.2.1 2
We preface the proof with the following lemma 3.2.13 Lemma: Let S be a countably separated Bore/ G-space and H c G a closed subgroup . Then H acts smoothly on S if and only if G acts smoothly on S x GjH. In particular, if H 1 , H 2 c G are two closed subgroups, then H 1 acts smoothly on G/H 2 if and only if H 2 acts smoothly on G/H 1 Proof: The second statement follows from the first, because the first implies that
both are equivalent to G acting smoothly on G/H 1
x
G/Hz. (Cf. 2. 2 2, 2 ..2.3 .) It
46
Ergodic theory and semisimple groups
is straightforward to see that the inclusion map S --" S x GjH, s --" (s, [e] ) defines a Bore! bijection i : S/H -* (S x GjH)jG. It suffices to show that the inverse map is also Bore!. Choose a Bore! section cp : G/H --" G of the natural projec tion p : G -" G/H with cp( [e] ) = e. Note that p(cp(xg)) = p(cp(x)g). Define )� : S x GjH --" S by A(s, x) = scp(x) - l Then one easily checks that A(sg, xg) = ).(s, x)h for some h E H, and hence ) induces a Bore! map (S x G/H)/G -" S/H which is clearly the inverse of i Proof (of Corollary 3.2.12): Let f1 E M ( IP" - 1 (k)}. Then w Gk c 1 1 PGL(n, k) and by
Theorem 3.2. 6 and 21 . 14, we can identify 11 PGL(n, k) with PGL(n, k)/PGL(n, k)w. Thus, it suffices to see that the Gk orbits on PGL(n, k)/PGL(n, k)11 are locally closed.. If PGL(n, k)11 were algebraic (e . g. , k = IR by Proposition 3.2.4), this would follow immediately from Theorem 3J.J . In general, by Proposition 32.4, we can find subgroups Hk c Q c PGL(n, k)11 where H is algebraic over k, both inclusions are as normal subgroups, and both quotient groups are compact By Lemma 32. 1 3, Gk is smooth on PGL(n, k)/PGL(n, k)11 if and only if PGL(n, k)11 is smooth on PGL(n, k)/Gk. Since Hk is smooth on PGL(n, k)/Gk, it clearly suffices to verify the following. Lemma: Suppose S is a countably separated G-space and H c G is a cocompact normal subgroup. If H acts smoothly on S, so does G.
3.2,14
G/H acts on the countably separated space SjH, and hence the result follows from Proposition 21 21. Proof:
Just a s Corollary 3.. 2 . 1 2 gives us information on the orbit space of M ( IPn - 1 ( k)) under a general Gk, the following result gives us information on the stabilizers .
Suppose G c SL(n, K) is a connected k-group and that the action of Gk on k" is irreducible Then for any measure f.lE M(IP" - 1 (k) ), we have either (i) ( Gk)11 is compact; or (ii) there is an algebraic k-group L c G with dim L < dim G and Lk => (Gk)w 3.2.15
Proposition:
Proof: If (Gk)11 is not compact, we can choose V; as in Corollary 32.2. Let M be the algebraic k-group with Mk = { g E SL(n, k) l g(u V;) = u V; } , and L = G n M. By construction, (Gk}11 c Lk. If dim G = dim L, then L = G by connectedness, and so G c M and hence
47
Algebraic groups and measure theory
G
c
M 0 Therefore, Gk leaves each V; invariant, contradicting irreducibility.
We can also generalize the results we have obtained so far concerning the action of Gk on M(IP" - 1 (k) ) to actions on the space of measures on more general varieties Definition: Suppose G is an algebraic k-group, and H c G is a k-subgroup We call H k-cocompact in G if Gk/Hk is compact
3.2.16
For example, all the Grassmann spaces and the various flag spaces over k arise as Gk/Hk where Gk = SL(n, k).. We also recall that if G is an algebraic group, an algebraic subgroup H c G is called a parabolic subgroup if G/H is a projective variety. If G is a k-group and H is a parabolic k-subgroup, then H is k-cocompact By Chevalley's theorem (3. 1 4), there is a r ational representation of G defined over k such that Hk is the stabilizer in Gk of a point x E IP" - 1 (k). Thus, we can identify Gk/Hk as a closed subset of IP" - 1 (k), and hence we also have an identification of M(Gk/Hk) as a closed Gk-invariant subspace of M(IP" - 1 (k)). 3.2.1 7
Corollary [Zimmer 4]: If H
is k-cocompact in G, then Gk acts smoothly on
M(Gk/Hk} Corollary: Suppose k lP and H c G is IP-cocompact. Let flE M(GIR/HrJ Then (GIR),u is the set of real points of an algebraic IP-group.
3.2.1 8
=
When H is a minimal IP-parabolic subgroup, 3.2.1 8 is due to [Moore 2]. Corollary: Suppose G is a connected k-group, almost simple over k, and that H c G is a proper k-cocompact subgroup. Then for any measure f1 E M(Gk/Hk), 3.2.19
either
(i) (Gk).u is compact; or (ii) there is an algebraic k-group L c G with dim L < dim G and Lk :::> (Gd.uProof: Identify Gk/Hk as an orbit in n:
IP"- 1 (k) for some rational representation
Gk ---+ GL(n, k). We may clearly assume that the orbit Gk/Hk linearly spans k".
Since n is a rational representation, n: Gk/ker(n) ---+ n(Gk) is an isomorphism (of topological groups) onto a topologically closed subgroup of GL(n, k) and since G is almost k-simple, ker(n) is finite. Thus, (Gk).u is compact if and only if its
48
Ergodic theory and semisimple groups
image in PGL(n, k) is compact If [n(Gk)]11 is not compact, we argue as in 3 . 2. 1 5 Namely, choose Vi as in Corollary 3 . 2. 2, let M be the k-group with Mk = {gE GL(n, k) l g(u Vi) = u Vi } , and L = n - 1 (M).. By construction (Gk)11 c Lk, and hence it suffices to see that dim L < dim G. If they were equal then by con nectedness of G, L = G and again by connectedness, n(Gk) leaves each Vi in variant Since Jl( [ Vi] ) > 0, (Gk/ Hk) n [ Vi ] # 0, which by invariance of Vi implies Gk/ Hk c [ Vi] This contradicts the assumption that Gk/Hk linearly spans k". It is natural to enquire as to the relation of the algebraic groups (G11)11 and H 11 in Corollary 3..2.1 8, and although we shall not actually require this informa tion in the sequel, we include some observations for completeness . The following result shows that the non-compact semisimple part of ( G11)11 can only be as large as the non-compact semisimple part of H11
With notation as in Corollary 3..21 8, let us further assume that L c G is an algebraic P--group with L11 c (G11)11 and L11 a semisimple Lie group with no compact factors . Then L� is contained in a conjugate of H 11
3.2.20 Theorem:
We begin the proof with the following useful general observation. 3.2.21
Lemma: Let G be a locally compact group acting continuously on a compact
metrizable space X. Let M0 = {J1 E M(X ) I J1 is G-invariant}, so that M o is a compact convex subset of M (X ) . Assume M o # 0 Then each extreme point of M o is ergodic under G . Proof: If J1 is not ergodic let A c X be G-invariant with A neither 11-null nor 11-conulL Let J1 1 (B ) = Jl(B n A)/Jl(A), Jlz(B ) = J1(B n (X - A) )/Jl(X - A).. Then Jl; E M0, f..li # Jl, J1 = Jl(A) J11 + f..l(X - A) Jlz, showing that J1 is not extreme.
L� leaves a measure invariant, by the lemma, there is an ergodic invariant measure v . However, L11 (and hence L�) acts smoothly on G11/H 11, and hence v is supported on an L�-orbit Thus, replacing H IR by some conjugate subgroup (in GIR) v is an L�-invariant measure on L�/ L� n H IR By the Bore! density theorem (3..2.5), L� n H IR = L�, completing the proof In particular, if HIR is solvable, or more generally, has a cocompact solvable normal subgroup, then H11 has no non-compact connected simple subgroups, and hence neither does (GIR)�< for any measure Jl. This implies from the structure Proof (of Theorem 3.2.20): Since
.
49
Algebraic groups and measure theory
theory of Lie groups that (Grrl)Jl has a cocompact solvable normal subgroup. For example, this will be the case for SL(n, IR) acting on the space of full flags in IR" More generally, if G is semisimple and P c G is a minimal parabolic IR-subgroup, then PR has a cocompact solvable normal subgroup Hence, we have the following, first obtained by [Moore 2] Corollary [M oore 2]: Let G be a semisimple IR-group and P c G a minimal IR-parabolic subgroup. If J.1E M(GIT1/PIT1), then (GITl)Jl is (the real points of) an algebraic group and has a cocompact solvable normal subgroup 3.2.22
Finally, we observe that since a connected semisimple Lie group of trivial center is of finite index in the IR-points of an algebraic IR-group (3.. L6) from Corollaries 3..2.1 7, 3 21 8, 3.2. 22 we easily deduce:
Let G be a connected semisimple Lie group with trivial center, and P c G a parabolic subgroup. (Recall for example, that if G = SL(n, IR) the parabolic subgroups are the stabilizers of flags.) Then every G-orbit in M(G/P) is locally closed and the stabilizers are algebraic. (Recall that we call M c G algebraic if, writing G = H � as in 3. 1 . 6, we have M = G n Lrr1 for some IR-group L c H) If P is a minimal parabolic, the stabilizers are compact extensions of solvable groups. 3.2.23
3.3
Corollary:
Orbits in function spaces
In the previous section we saw that the action of an algebraic group on certain (infinite dimensional) spaces of measures behaved in some respects like the action on a variety.. In this section we wish to observe the same phenomenon for actions on certain spaces of functions . We will actually be considering two different types of function spaces, the first being measurable functions into a variety. Let (X, J-1) be a finite measure space and suppose V is a complete separable metrizable space. Let F(X, V) be the space of measurable functions X -> V, two functions being identified if they agree almost everywhere . We recall that if [n, f E F(X, V), we say that fn -> f in measure if for every e > 0,
J.l {x I d([n(x), f(x)) � e } -> 0 as n --.
oo,
where d is a metric on
V
It is easy to see that this does not depend
50
Ergodic theory and semisimple groups
on the metric d, and depends only on the measure class of p, not on p itself If p is O"-finite, we say that fn -+f in measure if it does so with respect to one (and hence any) finite measure in the same measure class as p. This is easily seen to be equivalent to convergence in measure on all subsets of X of finite p-measure. If (X, p) is O"-finite, convergence in measure in F(X, V) defines a topology on F(X, V) which is metrizable by a complete separable metric. F urthermore, if 1, ---+ I in measure, then by passing to a subsequence we have pointwise convergence almost everywhere.. (The reader is referred to any standard book on measure theory for these assertions, e.g. [Berberian 1].) Suppose now that G is a locally compact group acting continuously on a locally compact V Then it is easy to check that G also acts continuously on F(X, V) via the action (f g)(x) = f(x) g, for I E F(X, V), g E G. If X is finite, this is of course just the product action of G on V x . . .. x V, and as such may or . may not be smooth. In particular, if G is the set of k-points of an algebraic k-group and V is the set of k-points of a k-variety on which G acts k-regularly, then for X finite, the action will be smooth by Theorem 1 1.3 (for k a local field of characteristic 0).. Margulis has observed that the action is still smooth for an arbitrary (O"-finite) measure space.
[Margulis 1 J Suppose G is a k-group, V a k-variety, and that G acts k-regularly on V.. Let G = G k , V = Vk , and assume k is a local field of characteristic 0. Let X be a (standard O"-jinite) measure space. Then the action of G on F(X, V) is smooth and the stabilizers are k-points of algebraic k-groups. 3.3.1
Proposition: .
8]: By Theorem 2. 1.14 (or more easily by Lemma 21 15), it suffices to show that for each I E F(X, V), G/G 1 -+I G is a homeomorphism. That is, it suffices to show that if/ g, -+f, then [gn ] ---+ [e] in G/G 1 . For this it suffices to show that each subsequence of [gn ] has a sub-subsequence converging to [e], and hence it suffices to consider the case in which (f gn )( x) -+f(x) on a conull set X 1 c X It is clear that g E G1 if and only if g E G 1 (x) for almost all x E X Let H 1 be a countable dense subgroup of G1 . Then for some conull set Proof [Zimmer
.
X 0 c X 1, we have HI c G1 c
n
xeXa
n
xeX0
G I (X)·· Since the latter group is closed, we have
G l <xl and since the reverse inclusion is clear, we have G1
=
n
xeXo
GHxl ·
The groups G 1 <xl are all algebraic over k since G acts k-regularly on V, and by the descending chain condition on algebraic subgroups, there exists x 1 , . . . , Xn E X 0 such that G 1
=
n
n Gt <x,)
i= 1
By Theorem 3. L3, the action of G on
n
0 GjG1<x,) is
' i =- 1
51
Algebraic groups and measure theory
smooth, and hence by 2. 1 . 1 4, the map G/G([eJ, , [e]} -+ ([e], morphism But clearly G([eJ, . [elJ
=
n G t <x,1, ;
. , [e]) G is a homeo
and hence we have a homeo
morphism
GfG t -+ ([e], . , [e]) G c IT G/G r < x ,J ·
Therefore, to see that [g11] --+ [ e] in G/G 1 , it suffices to see [gn] --+ [ e] in each G/G f(x,J· This follows directly from Theorem 21 . 14 and the fact that G acts smoothly on V The second type of action on a function space we will consider is an action on the space of rational functions between varieties . Namely, assume now that V is a (complex) variety defined over IR, such that VIR is Zariski dense in V, and let F(VIR, IP m(IR) ) be as above. Define Rat( VIR, IPm(IR)) = {f E F( VIR, IP m(IR)) If is the restriction to VIR of a rational function f: V --+ pm( C) defined over IR }. The result we will need is the following.
Let G be an algebraic IR-group acting on IP"(C) via a rational representation defined over IR, and suppose H and an H-action on !Pm(C) are similarly defined. Suppose V c lP"( C) is a closed G-invariant IR-subvariety, with VIR Zariski dense in V (In particular, GIR leaves the Lebesgue measure class on V11 invariant.) Define a GIR x H 11-action on F( V11, IP m(IR)) by (f (g, h) )(x) = f(xg - 1) h, so that Rat( V11, IPm (IR)) is an invariant set . Then (i) The G11, H 11, and G11 x H 11 actions on Rat( V11, IPm(IR)) are smooth. (ii) The stabilizers of points in Rat( V11, IPm(IR)) are real points of algebraic IR-groups. 3.3.2
Proposition:
Proof: The idea of the proof is that rational functions can be identified with a
class of closed subvarieties of a certain projective space, and that the space of subvarieties of projective space has the structure of a countable union of varieties Thus, the action in question can be reduced to a countable disjoint union of algebraic actions, for which the desired conclusions are known . We now sketch some further details. If X is a compact metric space, let �(X ) be the space of closed subsets, which is compact metric with the Hausdorff metric. (See e. g., the paragraph following the statement of Theorem 3 2. 6.) If X is a complex projective variety, we let Var(X ) c �(X ) be the set of closed subvarieties, and if X is defined over IR, let Var11(X ) be the closed IR-subvarieties. Let R( V, IPm(C)) be the rational maps V -+ Pm(C) defined over IR. For/E R( V, IP m(C)), the set of points at whichfis not defined is a closed IR-subvariety of IP"(C), which we denote by W1 . Hence f is regular on the quasi-projective variety V - W1 We recall that this implies that the
52
Ergodic theory and semisimple groups
graph of f, gr( f ) c (V - W 1) x !Pm(q is closed. (See, e.g. [Shafarevich 1 , p . 45].) Hence, if we let Z1 c P"(IC) x !Pm(q be the Zariski closure of gr(f), the map R( V, !Pm(C)) -+ Varrr1(IP"(C)) x Varrr1(1Pn(C) x !Pm(C)) given byf-+ ( Wt , Z 1) is injec tive. Give R( V, !Pm( C) ) the subspace topology for the topology defined by the Hausdorff metric on c&'(!Pm(C)) x c&'(IP"(C) x !Pm(C)). We also have the natural restriction mapping R( V, !Pm(C) ) -+ Rat( Vrr1, IPm(IR) ), which is injective since Vp is Zariski dense in V (and surjective by definition).. It is easy to check that this map is continuous, and hence a Borel isomorphism (Appendix A). Thus, it suffices to see that our results are true for the actions on Varp(IPn(C)) x Varp(IP n(IC) x IPm(C)) We can choose an isomorphism of IR-varieties of IPn(q x IPn(q x IPm(q with a closed IR-subvariety of a higher dimensional projective space lP'(C) in such a way that G x H has a compatibie rational representation into GL(r + 1, C) defined over R Therefore, it suffices to verify our results for the Gp x HR action on Varp(IP'(IC)). The stabilizers are clearly IR-algebraic, and hence only smoothness is at issue. However, it is well known, using the "Chow coordinates" of a projective variety (see e. g [Shafarevich 1], or [Samuel 1]) that there is a disjoint decomposition Var(IP'(C)) = u Y; into a countable union of GL(r + 1, q invariant Borel sets such that each Y; is a subspace of a projective IR-variety W; on which GL(r + 1, C) acts IR-regularly. Furthermore Y; n Varp(IP'(C) ) is con tained in ( W;)p. To see smoothness of Gp, Hrr1, Gp x HR on Varp(IP'(IC)), it suffices to see smoothness on ( Wi)p, which in turn follows horn Theorem 3 L 3..
3.4
Rationality of measurable mappings - first results
In this section we consider a different question relating measure theory and algebraic geometry, namely the question as to when one can deduce that a measurable map is r ational We begin by recalling one situation in which measurability implies a much stronger regularity property, namely in the case of homomorphisms of groups. More precisely, if n: G -+ H is a measurable homomorphism of second countable groups, and G is locally compact, then n is continuous. (For a proof, see Appendix B). Furthermore, if G, H are both Lie groups, it is well known that any continuous homomorphism is C 00. It is not true, however, that if G, H are (real points of) algebraic groups then the homomorphism must be rational. A simple example is the map IR -+ IR * , t -+ exp(t), where IR * is the multiplicative group of non-zero real numbers. The first result in this section is to describe a simple situation in which one can deduce that a homomorphism of algebraic groups is rational Let k be a field of charactistic 0.. A subgroup U c GL(n, k) is called unipotent if it is conjugate to a subgroup of N(n, k), where N(n, k) = {(aii) I au = 1 and
53
Algebraic groups and measure theory
a;j = 0 if i > j} A matrix A E G L(n, k) is called unipotent if all eigenvalues of A are equal to 1 . If U c GL(n, k) is the k-points of an algebraic k-group, then U
is unipotent if and only if every element of U is a unipotent matrix. (For a proof, see any of the standard references on algebraic groups.) For k = IR, a measurable representation of a unipotent group need not be rationaL F or example,
is a non-rational one-dimensional representation However, we do have the following
3.4.1 Proposition: Suppose k = IR or IC, and U is a k-group. Suppose U is uni potent. If n: Uk --+ GL(m, k) is a measurable unipotent representation, i.e., n(Uk) is a unipotent group, then n is k-regular
N(n, C) be defined as above, and let L(N) be the Lie algebra of N. That is, L(N ) is the linear subspace of all n x n matrices given by L(N ) = {(a;i) l a;i = O ifi � j} We remark that for any A E L(N ), A" = 0. The map
Proof: Let N =
n- 1
exp : L(N ) -+ N given by exp(A) = l: A ijj! is a regular mapping, and has j= O
n- 1
an inverse log : N -+ L(N ) given by log(B) = l: ( - l }i(B - I)ijj! (Note that j=O
for B E N, (B - I)" = 0..) Thus, exp is a biregular IR-isomorphism of varieties . If U c N is an (IR-) algebraic subgroup, and L(U) c L(N ) is the Lie algebra of U, then exp : L(U) --+ U will also be a biregular (IR-) isomorphism Since n is measurable, by our above remarks it is coo Let dn : L(Uk) --+ L(N(m, C)) be the derivative and, dnc : L(U) -+ L(N(m, C)) its extension to L(U ) (which only differs from L(Uk ) i f k = IR).. Then b y well-known properties of the exponential map, the following diagram commutes:
N(m, C) log
i
l
L(U)
dnc
�------""
exp
L(N(m, C) )
Since log, exp are regular, and dnc is linear, it follows that n is regular.
54
Ergodic theory and semisimple groups
In light of Proposition 3.41 , it is useful to have some criteria for determining when a representation of a unipotent group is unipotent In this direction, we have the following.
Proposition: Suppose G c GL(n, IR) is a semisimple Lie group and G � GL(m, IR) is a measurable (equivalently, C eo) representation If U c G is a connected unipotent subgroup, then n I U is a unipotent representation 3.4.2 n:
Proof: By letting ), = dn, it suffices to prove the following fact concerning Lie algebras: if f c yi(n, IR) is a semisimple Lie algebra, ,1. : f � yi(m, IR) is a representation, and A E f is a nil potent matrix, then A(A) E yi(m, IR) is also a nilpotent matrix. To see this, first recall that for any linear transformation A E End( V), V a finite-dimensional vector space, we can write A = A , + An in a unique way so that An is nilpotent, A, is semisimple (i.e. , is diagonalizable over an algebraically closed extension field), and A, and An commute. A basic fact about semisimple Lie algebras f is that if A E f, then A,, An E f. (For a proof� see [Jacobson 1] or [Bourbaki 1 ].) It is straightforward to check that if f c f i(n, IR) is a general Lie algebra and A E "' then A semisimple (resp nilpotent) implies ad(A): f � f is also semisimple (resp . nilpotent). If f is semisimple, the converse is also true. For if A = A , + An, then ad(A) = ad(A,) + ad(An) and by the preceding sentence, this is just the decomposition ad(A), + ad(A)n·· Thus, for example, if ad(A) is nilpotent, ad(A), = ad(A,) = 0, and since f is semisimple, ad is faithful and hence
A, = 0. With these observations, we now prove our required assertion. Since " is ) semisimple, so is the subalgebra ,1.(f c yi(m, IR). Thus, if A E f is a nilpotent matrix, to see ,1.(A) is as well, it suffices by the previous paragraph to see that adJo(A(A) ) is nilpotent However, the following diagram commutes:
f A
!
A( f
)
ad,.(A)
f
___ ad_,(A(A)) clearly is as well. We now turn to a different result, due to [Margulis 2]1 asserting rationality of certain measurable maps. This result will in fact be used in conjunction with
55
Algebraic groups and measure theory
34. 1 and 34. 2 in Chapters 5 and 6 to establish rationality of a specific measurable
map. Let V be a complex (IR-) variety, W an IR-variety, and A c WIR of positive (Lebesgue) measure. A measurable function / : A ---> V is called essen tially (IR-) rational if there is an (IR-) rational function R : W ---> V such that f = R a.e. on A .
3.4.3
Definition:
Theorem [Margulis 2]: Suppose f: IRk x IRn ---> V is a measurable function such that for almost all x E IR\.fx : IRn ---> V given by fx(y) = f(x, y) is essentially (IR-)' rational, and similarly, for almost all y E IRn, P : IRk ---> V given by fY(x) = f(x, y) is essentially (IR-) rational.. Then / is essentially (IR-) rational.
3.4.4
By induction, we can assume n = L For simplicity, let us first assume that V is affine . Then by taking coordinate functions, it suffices to assume V = C.. We first claim it suffices to prove the following: there is a set A c IRk of positive measure and a rational function R such that R = f a.e. on A x R For if we have this condition, then by Fubini's theorem, for almost all y E IR, RY = p on a set of positive measure in IR k . Since both RY and fY are essentially r ational for almost all y, this implies that for almost all y, RY = p a. e . on IRk Then Fubini's theorem implies R = f a. e. on IRk x IR. We now show that such a set A exists. For each pair of non-negative integers r, s, let A,,, c IRk be defined by A,,, = { x E IRk lfx : IR ---> IR is equal a . e . to a rational function Px/qx where degree(px) = r and degree(qx) = s}. Then A,,, are measurable sets and U A,,, is conull in IRk . Therefore, some A,,, has positive measure, Proof:
r,s
and we let A = A,,, for such a pair r, s. Then B = { y E IR l fx(y) = P x ( Y)/qx( Y) for almost all x E A,,,} is conull in IR by Fubini's theorem. Therefore, we may choose r + s + 1 distinct points A.0, . . . , A., + , E B with the further property that for each j, f;_; : IRk ---> IC is essentially rational. Set cj{x) = /A;(x) For x E A = A,,,, we can write r' - 1
s
P x( Y)/qx( y) = (y ' + L a;(x)y i)/ L b;(x) yi. i=O
i=O
Since f(x, y) = Px( Y)/qx( Y) a.e. o n A x IR, i t clearly suffices to see that a;(x), b;(x) are essentially rationaL However, a rational function pjq from IR into C with degree(p) = r, degree(q) = s, is determined by its values at any r + s + 1 distinct
Ergodic theory and semisimple groups
56
points. Therefore, for almost all x, a;(x), b;(x) are the unique solutions to the system of equations s -1 cj(x) = ( A.j + L a;(x)A.) )j L b;(x))��, 0 � j � r + s . r
i=O
i=O
For each x , this is a system of r + s + ! linear equations in the r + s + 1 unknowns a;( x), b;(x). Because c lx) is essentially rational in x for eachj, the coefficients ofthis linear system will be rational in x and hence so will its solutions (by Cramer's rule for inverting a matrix).. It follows that a ;(x), b;(x) are essentially rational, and by our observations above, so is f It is clear that [ will be essentially IR-rational if/x and f Y were assumed to be so . Finally, we reduce the case of a general V to an affine V Write V = u V; (finite union) where V; are affine and Zariski open. It clearly suffices to see that some f - 1 ( V;) is conulL Clearly, somef - 1 ( V;) is of positive measure. However, if y E IR such that fY is essentially rational and (fY) - 1 ( V;) is of positive measure, (fY)- 1 ( V;) must be conull in IRk A similar statement holds for fx, x E IRk A straightforward application of Fubini's theorem then shows that f - 1 ( V;) must be conulL
3.5
A homomorphism theorem
To describe the main result of this section, we begin with some general observa tions about abstract groups. Suppose A, B, C are groups with A c R If h : C --+ B is a homomorphism, then of course there is a map f: C --+ B/A such that f(xy) = f(x) h( y). There is also a type of converse assertion . Namely, suppose there is a map f: C --+ B/A and a map (not necessarily a homomorphism) h: C --+ B such that f(xy) = f(x)h(y). Let B0 c B be the subgroup that pointwise fixesf( C), and N(Bo) the normalizer of B0 in R Then for all y E C, h(y) E N(B0), and the induced map h : C --+ N(Bo)/Bo is a homomorphism. The group N(B0)/B0 acts on the fixed point set of B0 which contains /( C).. If we are given a map f: C --+ B/A , the existence o f such a map h: C --+ B can b e described a s follows. For each y E C, letfv : C --+ B/A beflx) = f(xy). Then the existence of such an h is equivalent to the statement that all fv, y E C, are in the same B-orbit in F( C, B/A) (the set of functions C --+ B/A). We summarize these remarks in the following:
Proposition: Let A, B, C be abstract groups, A c B, and let [E F(C, B/A). Suppose all fy, y E C are in the same B-orbit in F(C, B/A). Then there is (i) a subset W c B/A with f(C) c W, (ii) a group Q which acts on W; and
3.5.1
Algebraic groups and measure theory
57
(iii) a homomorphism h : C -+ Q, such that f(xy) = f(x) h(y) In particular, there is a point a E W such that f(y) = a h(y). The point of this section is to prove an analogous result where A, B, C are not abstract groups but are locally compact and all our hypotheses are assumed to hold only almost everywhere. In a sense, this is basically a technical result combining the observations of Proposition 1 5. L with a repeated use of Fubini's theorem. However, although it is technical, this result plays a basic role in the proof of the theorems of Chapters 5 and 6.. F(C, B/A) will now be taken in the sense of sections 13 and 34, namely functions are identified if they agree almost everywhere. The generalization of 15 1 we will require is the following
3.5.2 Proposition [Margulis 1]: Let C be a locally compact group, and k a local field of characteristic 0. Suppose H is an algebraic k-group, and L c H a k subgroup . Let cp E F(C, Hk/Lk) and for g E C, let (/Jg E F(C, Hk/Lk) be cp9(c) = cp(cg) Suppose that almost all (/Jg lie in a single Hk-orbit in F(C, Hk/Lk) ( This is equivalent to the existence of a map h : C -+ Hk such that cp(cg) = cp(c)h(g) for almost all (c, g). ) Then there exist (i) a k-subvariety W c H/L such that cp(c) E W for almost all c, (ii) an algebraic k-group Q which acts k-regularly on W; (iii) a measurable homomorphism h : C -+ Qk, and (iv) a point X E Wn Hk/Lk. such that cp(c) = x h(c) for almost all c E C. .
Proof: The essential range of cp : C -+ Hk/Lk will be a subset of HIL and we let B c H be the subgroup pointwise fixing the essential range. Then B is a k-group. Let W c H/L be the k-variety of all fixed points of B Let N(B) be the normalizer of B in H, so that N(B) leaves W invariant, and hence the algebraic k-group Q = N(B)/B acts k-regularly on the k-variety W Since almost all cpg lie in a single Hk-orbit, it follows that for each fixed a E C, we have (/Jag and cpg are in this Hk-orbit for almost all g E C. Thus, for any a E C, we can write (/Jag = cpg hg{a) for almost all g E C, where hg{a) E Hk and hg(a) is measurable in (g, a).. (To see this last measurability assertion, we use the existence of measurable sections (Appendix A).) The essential range of cpg for any g E C is identical to the essential range of cp, and it follows that for each a E C, hg(a) E N(B)k for almost all g Furthermore, we claim that for each a, b E C and almost all c, g E C, ..
58
Ergodic theory and semisimple groups
To see this, we simply observe that for each a, b and almost all c, g, q>g{c) h9(ab)
=
(/Jab9(c)
=
cp(cabg)
=
0.) We can consider cp as a map on the circle. If we let If; be a rotation of the circle by any angle which is not a multiple of 2n, then there are no probability measures on the circle invariant under both cp and If; . But cp, If; generate an F 2 action, showing non-amenability of F 2, from which the non-amenability of a general Fn follows. The following observation tells us when lattices in Lie groups are amenable. =
Proposition: Suppose r c G is a closed subgroup such that Gjr has a finite G-invariant measure. Then G is amenable if and only if r is amenable.
4.1.1 1
Proof: In light of 4.1 .6 it remains to show that r amenable implies G amenable
Let A be an affine G-space. Then there is a r-fixed point in A, say a0 Then g ---> a0g induces a G-map f: Gjr ---> A, and if J1 is the G-invariant measure on Gjr, thenf* (Jl) will be a G-invariant measure on A. Thus the barycenter b(f* (Jl)) will be a G-fixed point In particular, lattices in non-compact semisimple Lie groups are not amenable. The following result is usefuL
4.1 .12
Proposition: Every locally compact G has a unique maximal normal
amenable subgroup.
We begin with two easy lemmas. 4.1.13
Lemma: If H is amenable and cp : H ---> G a continuous homomorphism with
cp(H) dense in G, then G is amenable..
Ergodic theory and semisimple groups
64
Proof: If A is an affine G-space, it is also an affine H-space. Since cp(H) is dense,
an H-fixed point will be G-fixed. 4.1.14
Lemma: Suppose Ha c G where {Ha } is a directed set of amenable closed
subgroups Then uHa is amenable
Proof: This is a straightforward finite intersection argument
Proof (of 4. 1 12): By Lemma 4.1 . 14, there exists a maximal normal amenable subgroup. Suppose H h H 2 c G are two such groups.. Then H 1 acts by con jugation on H2 , so we can form the (external) semi-direct product H 1 1>< H 2 · This group is amenable by 4. 1 . 6. We have a continuous homomorphism H 1 1>< H 2 --+ H 1 H 2 c G, and hence H 1 H 2 is amenable by Lemma 4. 1 1 3 and is clearly normal. By maximality of H 1 , H 2 , we have Ht = H2. .
Perhaps surprisingly, the conclusion of Lemma 4. 1 1 3 is also true if G is the real points of an IP-algebraic group and we only assume cp(H) is Zariski dense 4.1.15
Theorem [Moore 2]: Suppose G is the set of real points of an IP-algebraic
group and cp : H --+ G is a (topologically) continuous homomorphism with cp(H) Zariski dense in G. If H is amenable, so is G.
Proof: It clearly suffices to consider cp : H --+ G
= GL(n, lP), and to show that cp(H)
(Zariski closure) is amenable. Let P be the subgroup of G consisting of upper triangular matrices so that G/P is the space of full flags. H acts on G/P and since H is amenable, it leaves a measure f1 E M(G/P) invariant However, from Corollary 3. 2.22, we see that Gll is both amenable and algebraic . Therefore, cp(H) c Gll is amenable by 4. L 6. We conclude this section by mentioning without proof (we will not be using this result) a striking theorem of J. Tits. To put this in some perspective, we recall that any discrete group containing the free group F2 cannot be amenable. A conjecture of von Neumann, which had remained open for many years, was that the converse is also 'true, namely that any non-amenable discrete group contains a non-abelian free group . A counterexample to this conjecture has recently been constructed by [Olshanski 1J However, Tits' result proves the von Neumann conjecture for linear groups. .
65
Amenability
Theorem [ Tits 3]: Suppose k is a field of characteristic 0, and r c GL(n, k) is a subgroup. Then either r contains a non-abelianfree group or r has a solvable subgroup of finite index (and hence is amenable).
4.1.16
4.2
Cocycles
Cocycles of a group action are a fundamental tool for understanding the action, and in this section we present an introduction to this important concept In the next section we will use cocycles to extend the notion of an amenable group to that of an amenable action. We shall be seeing many other uses as well. To motivate the discussion, suppose that (S, Jl.) is a G-space and that T is some countably separated Bore! space. As usual, we let F(S, T) be the space of measurable functions S -'* T, two functions being identified if they agree almost everywhere. Then G acts on F(S, T) by translation, i.e. (g f) = f(sg).. Suppose now that T is also a (Bore!) H-space for some other group H. (For notational convenience, assume H acts on the left.) Then we can define new actions of G on F(S, T) which consist of translation together with some "twisting" by H. More precisely, suppose that for each s E S, g E G, we have an element a(s, g) E H Then we define (g of)(s) = a(s, g)f(sg) - a translation and a twist by a(s, g) . For this to define an action, a must satisfy some compatibility conditions, namely: .
Definition: Suppose (S, Jl.) is a G-space and H is a second countable group. A Bore! function a : S x G -'* H is called a cocycle if for all g, h E G, a(s, gh) = a(s, g)a(sg, h) for almost all s E S. The cocycle is called strict if this equation holds for all s, g, h. 4.2.1
If a is a cocycle, then the above definition of g f yields an action of G on F(S, T) which we call the a-twisted action. Suppose now that a, f3 : S x G -'* H are both cocycles. What relation between a and f3 will ensure that the twisted actions on F(S, T) will be equivalent? 4.2.2
Definition: Two cocycles a, f3: S
x
G -'* H are called equivalent, or eo homologous, if there is a Bore! function cp : S -'* H such that for each g, f3(s, g) cp(s)a(s, g)cp(sg) - l for almost all s . We then write a "" {3. If a, f3 are strict, and the above equation holds for all s, g, we say that a and f3 are strictly equivalent =
.
.
If a "" {3, then the map : F(S, T) -'* F(S, T), given by (f)(s) = cp(s)f(s) is a Bore! isomorphism which is a G-map, where the domain has the a-twisted action and the range has the {3-twisted action .
66
Ergodic theory and semisimple groups
Pictorially, we think ofcocycles and cohomology as follows. Consider elements of F(S, T) as sections of the bundle S x T-+ S, and consider a(s, g) E H as giving a map from the fiber over sg to the fiber over s . In the twisted action, er:(s, g) then tells you how to bringf(sg) back to an element of the fiber over s .. Then the cocycle identity is simply commutativity of the following picture:
er: (s, g h )
�
er: (sg, h ) er: (s,:g) � � T
----�-� ----�-
sg
5
0------- -
sg h
s
The relation of cohomology is simply the existence of a bundle self--equivalence, i.e , for each s, a self-map cp(s) of the fiber, so that the following picture commutes:
er: (s, g )
[J (s, g)
�
�
-----cp (sg)
s
---
� sg
......____
Remarks: (1) By considering er: : S x G -+ H as an element of F(G, F(S, H)), we see that the cocycle identity is exactly the condition that a be a Borel 1-cocycle on G (in the sense of Eilenberg-MacLane) with values in the G-module F(S, H ),
67
Amenability
where the latter action is ordinary translation. Similarly, cohomology in the sense of Definition 4..2..2 is cohomology in the sense of Eilenberg-MacLane (using only Borel cochains). Although this point of view is quite useful for certain considerations, and r aises many interesting questions (e . g. concerning higher cohomology), we shall not pursue this connection here. See [Feldman Moore 1], [Zimmer 1 4]. (2) The question as to how close an arbitrary cocycle is to a strict cocycle is a technical point which is of considerable importance for certain considerations Although we will be using cocycles defined on an arbitrary G-space, the only actions for which we will need to deal with the above question are transitive actions . Thus, we will deal with this point for transitive actions in this section, and discuss the more technical general situation in Appendix B. We remark, however, that if G is countable, it is easy to see that for any a, there is a strict cocycle a' such that for all g, a(s, g) = a'(s, g) a. e. Namely, for g, hE G, let Ag.h = {sla(s, gh) = a(s, -
g)a(sg, h)}, A = n Ag.h and So = n A · g. Then So is an invariant conull Borel set, ge G g. h and hence we can take a'(s, g) = a(s, g) for all sESo, a'(s, g) = e for srfS0 We now consider some cocycles that arise naturally.
4.2.3 Example: Suppose M is a manifold of dimension n and G acts on M by diffeomorphisms. For each s E M, g E G, let a(s, g) = (dg - 1 ),g : TM"' -+ TM,, where TM is the tangent bundle. Then the chain rule for differentiation is exactly the cocycle identity of Definition 4.2. L We can always choose enough Borel sections of the tangent bundle, and hence measurably TM is triviaL Under any trivializa tion, a will be identified with a strict cocycle M x G -+ GL(n, IR). Different trivializations of TM will yield strictly equivalent cocycles. The a-twisted action is just the natural action of G on measurable vector fields on M.
4.2.4 Example: Suppose (S, Jl) is a G-space with quasi-invariant measure fl· Let r" : S x G -+ IR + be defined by r"(s, g) = dJ1(sg)/dJ1(s), the Radon-Nikodym deriva tive. Once again, the chain rule is exactly the cocycle identity and r" is called the Radon-Nikodym cocycle. If J1 and v are equivalent measures, so that dfl = cpdv, cp > 0, then r"(s, g) = cp(sg)dv(sg)jcp(s)dv{s), ie , cp(s)rtls, g)cp(sg) 1 = rv(s, g).. Thus, r" "' rv. Conversely, any cocycle r : S x G -+ IR + with r" � r i s of the form rv, where v is a measure equivalent to J1 Since rv = 1 (a.e. .) if and only if v is G-invariant, we have:
68
Ergodic theory and semisimple groups
4.2.5 Proposition: If (S, Jl) is a G-space, then there is a ((J·finite) G.-invariant
measure equivalent to J1 if and only if the Radon-Nikodym cocycle r11 is equivalent to the identity cocycle, i e , i(s, g) = 1 for all s, g
4.2.6 Example: Suppose n : G -+ H is a (continuous) homomorphism. Then a,. : S x G -+ H given by a,.(s, g) = n(g) is a cocycle. The cocycles of the form a,. are exactly those that are independent of s. An important example of a cocycle is one coming from an orbit equivalence of actions. 4.2.7 Definition: Suppose (S, /1) is a G-space, (S', 11') a G'-space. The actions are
called orbit equivalent if there are conull Bore! sets S0 c S, S� c S' and a measure class preserving Bore! isomorphism @ : S0 -+ S� such that if s, t E So, then s and t are in the same G-orbit if and only if O(s) and O(t) are in the same G'-orbit
4.2.8 Example: IfO is an orbit equivalence, and the action ofG' on S' is essentially free (i..e . , almost all stabilizers are trivial), then for each g and almost all s, there is a unique a(s, g) E G' such that O(s)a(s, g) = O(sg).. It is straightforward to check that a is a cocycle . Some of the significance of its cohomology class can be seen from the following.
4.2.9 Proposition: Suppose G = G' in Example 4..2..8 . As above, let a : S x G -+ G .
be the cocycle corresponding to an orbit equivalence of essentiallyfree G-spaces, O : S -+ S' If there is an inner automorphism n : G -+ G such that a � a,, (the latter as in Example 4. 2 . 6), then the actions of G on S and S' are isomorphic.. .
We shall indicate the idea of the proof, which in fact is essentially a complete proof if G is discrete. A full treatment in the general case becomes somewhat technical and we shall refer the reader to [Zimmer 9] for details. Proof: Suppose that for each g, a(s, g) = cp(s)a,(s, g) G' is an isomorphism then identifying G and G' by this isomorphism, the actions of G on S and S' will be isomorphic 4.2. 1 1
4.2.1 2
Example: Cocycles of Z-actions can be easily described Namely, if S is
a Z-space, and ex : S x Z -> H is a cocycle, let f(s) = ex(s, 1). Then by repeated applications of the cocycle identity, ex(s, n) can be expressed in terms of f for any n. Conversely, any function f : S -> H can arise this way. Thus cocycles on Z-spaces can be identified with Bore! functions S -> H We now consider cocycles on transitive G-spaces. Suppose Go c G is a closed subgroup and that ex: GjG0 x G -> H is a strict cocycle. Then the cocycle identity immediately implies that (Ja : G0 -> H given by (Ja(g) = ex([e], g) for g E G0 is a Bore! (and hence continuous) homomorphism. Furthermore, if ex, f3 are strict cocycles and ex and f3 are strictly equivalent, then (Ja and (Jp are conjugate homomorphisms, i .e. , there is h E H such that (Ja(g) = h(Jp(g)h - 1 We now make the following simple but important observation: Every (measurable) homo morphism (J ; G0 -> H is of the form (Ja for some strict cocycle ex: G/Go x G -> H To see this, we first observe that it is true for the identity homomorphism i : G0 -> Go. Namely, choose a Bore! section y: G/Go -> G of the natural projection with y([e]) = e. Then for x E G/G0 and g E G, y(x)g and y(xg) are equal when projected to G/G0 . Therefore, f3(x, g) = y(x)gy(xg) - 1 E G0, and it is immediate that this is a strict cocycle, and that (Jp = i . If (J : Go -> H is any homomorphism, then ex(x, g) = (J(y(x)gy(xg) - 1 ) is a strict cocycle with (Ja = (J We summarize our discussion in the following. .
70
Ergodic theory and semisimple groups
4.2.13 Proposition: The map {Strict cocycles G/Go x G -+ H } - -+ Hom( G 0 , H), a - -+ O"a, is a surjection which induces a bijection between strict equivalence classes of strict cocycles and conjugacy classes of homomorphisms . Furthermore, any a is strictly equivalent to some a' such that a'( G/ G o x G ) = a-a( Go ) .
Proof: In light of the above discussion, all that remains to be verified is that O"a
and a-13 conjugate implies a and f3 are strictly equivalent We first observe that from the cocycle identity, letting y be a section as above, that a([e], y(x) )a(x, g)a([e], y(xg)) - 1 = a([e], y(x)gy(xg) - 1)
=
a-a(y(x)gy(xg) - 1 ),
and that we have a similar formula for [3.. If a-a ( ) = h - 1 a-13( )h, then we imme diately have a(x, g) = cp(x)f3(x, g)(p(xg) - J
where cp(x) = a([e], y(x) ) - 1 hf3([e], y(x)}.
As an illustration we present the following:
4.2.14 Example: Consider O(n) (the orthogonal group) acting on sn - 1 � O(n)/ O(n - 1).. As in Example 4..2.3, we have the derivative (strict) cocycle
a : S " - 1 x O(n) --+ G L(n - 1). By the above proposition; this corresponds to a homomorphism O(n - 1) --+ G L(n - 1), and it is easy to verify from the construc tion of a that the homomorphism is just the usual inclusion map. For transitive G-spaces, the study of arbitrary cocycles can be reduced to that of strict cocycles .
4.2.15 Proposition [Mackey 1]: Consider cocycles on the G-space G/G0, where G0 c G is closed. Then (i) If r:t. is any cocycle, there is a strict cocycle r:t.' such thatfor each g, r:t.(s, g) a. e. (ii) If two strict cocycles are equivalent, they are strictly equivalent.
=
r:t.'(s, g)
Amenability
71
Proof: (i) I f G 1 i s a conjugate o f G o i n G, then we have a n identification o f G
spaces G/G0 � G/G 1 , and hence we are free to replace Go by a conjugate. Ifcx is a eo cycle, then by Fubini we have that for almost all x E G/G0, cx(x, gh) = cx(x, g)cx(xg, h) for almost all (g, h) E G x G. Replacing Go by a conjugate if necessary, we can assume that this holds for x = [e] Thus cx([e], gh) = cx([e], g)cx([e]g, h) for almost all g, h, and so for any a E Go cx([e], agh) = cx([e], ag)cx([e]g, h) a.e.
Let !J(g) = cx([e], g). Then from the above two equations, we have that for each a E Go,
for almost all g, h. Thus, for each a, the function g __,. !J(ag)�J(g) - 1 considered as an element of F(G, H) is invariant under translation by almost all elements h E G, and since a eonull subgroup cannot be proper (Appendix B) this is invariant under all G. By Lemma 2.2.. 16, for each a, there is a (unique) element n(a) E H such that for each a E G0 and almost all g E G, !J(ag)!J(g) - 1 = n(a).. The last equation clearly implies that n is a homomorphism. (To see that n is Bore!, embed H as a Borel space in [0, 1], and note that n(a) = J !J(ag)!J(g) - 1dg, where dg is a probability measure in the class of Haar measure . ) Let [3 be a strict cocycle corresponding to the homomorphism n : G0 __,. H, and let A.(g) = [3([e] , g). Then for a E Go A.(ag)A.(g)- 1 = n(a) = !J(ag)!J(g) - 1 for almost all g, i.e , !J(g) - 1 A.(g) = !J(ag) - 1 A.(ag). Once again by Lemma 2..2J 6, we can find
H, and T is a left H-space . A function f : S --> T will be called a-invariant if it is a fixed point for the a-twisted action of G on F(S, T). Equivalently, for each g, a(s, g)f(sg) = f(s) a.e. 4.2.18 Example: (a) If T = H, then there is an a-invariant function S --> H if and
only if a is trivial, i.e., equivalent to the identity cocycle. (b) More generally, if T = H/H0, then there is an a-invariant function S --> H/Ho if and only if a � f3 where fJ(S x G) c H0 For if cp is a-invariant, then for each g, fJ(s, g) = cp(s) - 1a(s, g)cp(sg) E Ho a.e., and by changing f3 on a null set, we have fJ(S x G) c H 0. Conversely, if a � f3 with f3 taking values in H 0, and cp implements the equivalence, then cp - 1 will be an a-invariant function (c) If a "' /3, then there is an a-invariant function S --> T if and only if there is a /]-invariant function.
73
Amenability
If G is transitive, we can relate a-invariant functions to fixed points of homo morphisms corresponding to a
4.2.19 Proposition: Suppose a : G/G0
x G ---> H is a strict cocycle and a : Go ---> H the corresponding homomorphism. If T is an H-space, then there is an a-invariant function G/G0 ---> T if and only if there is a fi xed point in T for the action of a(Go).
Proof: By 42.1 3 we can replace a by a strict cocycle whose values lie in a(G0)
It is then clear that if t E T is a(Go) invariant, cp(s) = t for all s will be a-invariant To see the converse, let us first extend a to a map G ---> H by setting a(g) = a([e ], g). Then a I Go is a homomorphism, but moreover a(hg) = a(h)a(g) for h E G0 and g E G We can recover a from a via the cocycle identity. Namely, a([x], g) = a([e] x, g) = a([e] , x) - 1 a([e], xg), so a([x], g) = o'(x) - 1 a(xg). Thus, if cp is a-invariant, we have that for each g and almost all x, a(xg)cp([x] g) = a(x)cp([x]). By Fubini, we can choose [x] E G/Go such that this holds for almost all g E G, and hence a(y)cp([y]) = a(x)cp([x]) for almost all y E G .. Set a(x)cp([x]) = a E T Then for h E G0, tT(hy)cp([hy]) = a for a. e. y E G, ie , tT(h)tT(y)cp([y]) = a for a e y e:: G . As tT(y)cp([y]) = a a e , we deduce tT(h)a = a for all h e:: G0
Remark: The above proof shows not only that there is a G0-fixed point, but that
the fixed point a E T can be chosen such that for almost all g E G, tT(g)cp([g]) = a, where tT(g) = a( [e], g). We remark that if we are dealing with Borel functions themselves rather than function spaces F(S, T) where functions are identified if they agree a.e., and the a-invariance held everywhere, not just a.e., then 4 . 2.. 1 9 would just be a matter of formal algebraic manipulation. However, function spaces like F(S, T) are in general much better behaved than the space of Borel functions, and in particular, it is much easier to deduce the existence qf fixed points from general functional analytic arguments. Thus the necessity for the a. e. argument in the above proof. If the space T is a linear space, and H acts on T by linear transformations, then F(S, T) will also be a linear space and the a-twisted action of G will be a linear action. In particular, if S = G/G0, and we have a linear representation of Go on T, then we have an associated cocycle and hence an associated twisted linear action . Thus, we pass from a linear representation of G0 on a space E to a linear representation of G on F(G/G0, E ). This is just the classical notion of an induced representation. If the original representation has further structure, .
74
Ergodic theory and semisimple groups
e.g.. is an isometric representation on a Banach space, or a unitary representation on a Hilbert space, we would like the induced representation to have the same structure, and we achieve this by passing to a suitable subspace. More precisely, suppose E is a separable Banach space, Iso(E) the group of isometric auto morphisms of E, and that 0': G0 --> I so(E) is a continuous representation (where Iso(E ) has the strong operator topology). Assume for the moment that G/Go has a 0'-finite invariant measure . Then we have an induced continuous represen tation of G by isometries on the Banach space L 1 (G/G0,E) c F(G/G0,E) given by (re(g)f)(x) = cx(x, g)f(xg) where ex is a cocycle corresponding to 0' The represen tation is by isometries because cx(x, g) E Iso(E) and G preserves the measure on G/G0. The representation is easily seen to be measurable, and hence it is continuous (Appendix B).. If G does not have an invariant measure on G/G0, let p: G/Go x G --> IR + be the Radon-Nikodym derivative for a quasi-invariant measure. Then we define the induced representation of G on L 1 ( G/G0,E) by (re(g)f)(x) = p(x, g)cx(x, g)f(xg). (We remark that this makes sense for any cocycle ex taking values in Iso(E) defined on any G space . ) We remark that whether or not there is an invariant measure, the formula for the adjoint representation re* of G on e(G/G0, E)* � L 00(G/G0, E * ), namely re*(g) = (re(g) - 1)*, is given by (re*(g)
sg is a measure class preserving Borel isomorphism S -> S, but for g, h E G, we only have s gh = (sg) h a . e. (Actually, every near action is "equal almost everywhere" to an action . See Appendix B. ) In the case of inducing, we formed the space of H-orbits (S x G)/H and let G act on this space. However, for a general strict cocycle a, the space of orbits may be badly behaved, and one may even have that H acts properly ergodically on S x " G. We thus require a replacement for the space of H-orbits. Let I = {!E L co (S x " G) I for each h, f(zh) = f(z) a e } Then I is easily seen to be a weak-*-closed subalgebra of L 00 (S x G), and hence there is a standard measure space (X, v) and a measure class preserving Borel map cp : S x G -> X such that cp*(L 00 (X )) = I c L 00(S x G). If (S x " G)/H is countably separated, then we can naturally identify X � (S x " G)/H (We remark that X and cp are unique up to null sets.) As in the inducing pr ocess, G acts on S x " G by (s, g)g0 = (s, g0 1 g),
Amenability
77
and this commutes with the near action of H. Therefore, G leaves the algebra I invariant, and hence (see Appendix B) we may choose X to be a G-space and qJ to be a G-map. 4.2.23
Definition: The G-space (X, v) is called the Mackey range of the cocycle x
The action was introduced in [Mackey 5] We list some properties of the Mackey range . Once again, as we will not be using them in the present work, we omit the proofs, but invite the reader to supply them (For non-discrete groups, the technical results of Appendix B should be useful Some detailed proofs can be found in [Rap1say 1 ] and [Zimmer 1].)
4.2.24
Proposition [Mackey 5]
(a) The Mackey range action is always ergodic (assuming the original H-action is ergodic). (b) If the skew product action of H on S x a G is ergodic, the Mackey range is the action of G on a point. (c) The Mackey range is (up to isomorphism) a cohomology invariant of x . (d) The cocycle is trivial (i. e. , equivalent to i(s, g) = e) if and only if the M ackey range is the action of G on itself by translation (e) a is equivalent to a cocycle taking values in a subgroup G o c G if and only if the M ackey range is induced from an action of Go (f) IfS = H/Ho and x : H/Ho x H -> G corresponds to a homomorphism (J : Ho -> G, then the Mackey range is the action of G on G/(J(Ho).
Finally, we mention one further example. Suppose S; is an essentially free ergodic G; space, i = 1, 2, and that the actions are orbit equivalent Let a : S 1 x G 1 -> G 2 be a cocycle corresponding to an orbit equivalence Then the Mackey range of a is the G 2 action on S 2 4.3
Amenable actions
If G is an amenable group, any action of G will inherit certain proper ties from the group. On the other hand, it is possible that a given action of a non-amenable group may also have these properties . This leads us to the notion of an amenable action of a group, first introduced in [Zimmer 3} This notion has proven useful in a variety of situations. In particular, we shall see that if r c SL(2, IR) is a lattice, then the action of r on the boundary IR (see introduction) is amenable, in spite of the fact that r is not More generally, if G is a Lie group and r c G
78
Ergodic theory and semisimple groups
is a lattice, then r acts amenably on G/H if H c G is an amenable group. In particular, this will be the case if G is semisimple and H is a minimal parabolic subgroup, thus generalizing the example above of r acting on IR . The results on amenable actions we present here follow [Zimmer 3, 4] . Amenability o f G demands the existence of a fixed point in every affine G-space Amenability of an action of G on a space (S, Jl) will be the condition that G has a fixed point in certain affine G-spaces which are nicely related to S To describe these affine actions, suppose E is a separable Banach space and a : S x G --. Iso(E) is a cocycle. We wish to describe certain G-invariant compact convex sets in L 00(S, E*), where G acts on the latter by the adjoint of the a-twisted action on L1(S, E), namely (n*(g) G2 the corresponding cocycle [Example 42 8] Since G 1 is discrete, we can assume, by passing to a conull subset, that a is strict If f3 : S2 x G2 -> Iso(E) is a cocycle, and F(S2, {Ar}) an affine Gz space over Sz, then P : S 1 x G 1 -> lso(E), fJ(s, g) = f3(8(s), a(s, g)) is a cocycle and F(S 1 , {A9(sJ}) is an affine G 1 -space over s 1 If cp E F(S 1 , {A9(s)}) is a P-invariant function, define ljl(t) = cp(8 - 1 (t)). For each h E H, and almost all t E S2, we can write (t, h) = (8(s), a(s, g) ) for some g E G 1 , and since G; is countable, it follows that ljJ E F(S2, {Ar} ) is a {3-invariant function . Thus, if the G 1 action is amenable, so is the G2 action
4.3.1 1 Corollary: Let S; be an essentiallyfree G;-space withfinite invariant measure, for i = 1 , 2. Suppose the two actions are orbit equivalent Then G 1 is amenable if and only if Gz is amenable.
Proof: 4..3.3 and 4.3.10.
The first major result concerning orbit equivalence was the fundamental theorem of H. Dye. Let Z be the group of integers. Theorem [Dye 1 , 2]: All .finite measure preserving (properly) ergodic Z-actions are orbit equivalent .
4.3.1 2
This was later extended to the a-finite case by Krieger Theorem [Krieger 2]: A ll a-finite (but not finite) measure preserving Z-actions are orbit equivalent.
4.3.1 3
Krieger also extended the theorem to the case of quasi-invariant measure
Amenability
83
without equivalent invariant measure. He showed that a measurement of the extent to which the action fails to be measure preserving is a complete invariant of orbit equivalence. Namely, let X be an ergodic G-space, and r:X x G -> IR + be the Radon-Nikodym cocycle. Let L1 : G -> IR + be the modular function of G, and let m : X x G -> IR + be m(x, g) = r(x, g)Ll(g) - 1 We call m the modular cocycle The Mackey range of this cocycle (Definition 4.2.23) will be an ergodic IR + -action, which we call the modular flow or the modular r ange . For a unimodular group G, the modular flow will be translation of IR + on IR + itself if and only if the Radon-Nikodym cocycle is trivial (Proposition 4.2. 24), and hence if and only if there is an equivalent invariant measure for the action (4.2..5).
Theorem [Krieger 2]: For Z-actions with quasi-invariant measure, and not possessing (an equivalent) finite invariant measure, the modular flow is a complete invariant of orbit equivalence.
4.3.14
Accessible expositions of Krieger's work are [Sutherland 1] and [Hamachi and Osikawa 1 ]
4.3.1 5
Example: Let u s see how t o compute the modular fl o w i n some examples. Let r c SL(2, IR) be a lattice, and let P be the upper triangular subgroup. As is well known from Lie theory (e. g.. [Helgason 1]), the Radon-Nikodym cocycle for the action of G = SL(2, IR) on G/P is the cocycle a : G/P x G -+ IR + corres ponding to the homomorphism P -> IR + given by Llp, the modular function of P. Clearly r : G/P X r -> IR + is just a l GjP X r. Now the r action on G/P X ,. IR + that appears in the construction of the Mackey range is just the restriction to r of the G action on G/p X a IR + . Since Ll : p -+ IR + is surjective, as is well known, this is the action of G on G/P x a Pjker Llp which is transitive with stabilizer kerllp Thus as a r-space, G/P X r IR + is just the action of r on G/Ker Llp . Since ker Llp is not compact, by Moore's ergodicity theorem (Chapter 2) r is ergodic on this space . Hence the modular flow is the action of IR + on a point. This computation can clearly be carried out on any simple non-compact Lie group. If r is a lattice in such a group G, and P c G is a minimal parabolic, then the modular flow of the action of r on G/P will be the action of IR + on a point. The Dye-Krieger theorems were extended over the years by a number of persons to include within its framework actions of larger classes of groups.. (In fact Dye did not restrict himself to the integers.) This work has recently culminated with the following theorems.
Frgodic theory and semisimple groups
84
[Connes-Feldman- Weiss 1]: (i) A free properly ergodic action of a discrete group is amenable if and only if it is orbit equivalent to a Z-action (ii) The Dye-Kr ieger theorems (4. 3 1 2-4.3 . 1 4) hold for the class of amenable properly ergodic actions of discrete groups 4.3.16
Theorem
4.3. 1 7 Theorem [Connes-Feldman- Weiss 1]: (i) A free properly ergodic action of a continuous group is amenable if and only if it is orbit equivalent to an !P-action (ii) For such actions, the modular flow is a complete invariant of orbit equivalence. In particular, any two free properly ergodic actions of continuous amenable unimodular groups with invariant measure are orbit equivalent
Example: If G is a simple noncompact Lie group, r c G at lattice, P c G a minimal parabolic, we saw in Example 4.JJ 5 that the modular flow of r on G/P is independent of G and r. Since these actions are amenable, Theorem 4. 3. 1 6 implies that they are all orbit equivalent For some further results on amenable actions see section 9.2
4.3.1 8
5
5.1
Rigidity
Margulis' superrigidity theorem and the Mostow-Margulis rigidity theorem
In this section we prove the rigidity theorems for lattices in semisimple Lie groups, making basic use of the results we have established in the preceding three chapters . Most of the results on semisimple groups we have obtained so far hold for an arbitrary semisimple group. In this section, we shall often require the assumption that IR-rank(G) � 2 . We recall that if G is a semisimple algebraic group defined over k, then k-rank(G) (or sometimes, by abuse of notation, k- rank( Gk) ) is defined to be the maximal dimension of an abelian k-subgroup of G which is k-split, i e , which can be diagonalized over k. If G is a connected semisimple Lie group then we can realize Ad(G) as a subgroup of finite index in the IR-points of an IR-group (Proposition 3. 1 . 6).. We then define IR-rank(G) to be the IR-rank of this algebraic group. Thus IR-rank(SL(n, IR) ) = n - 1, the IR-split abelian subgroup of maximal dimension being the diagonal matrices of determinant one .
5.1.1 Theorem (Mostow-Margulis Rigidity Theorem): Let G, G' be connected semisimple Lie groups with trivial center and no compact factors and suppose f c G and f ' c G' are lattices . Assume f is irreducible and IR-rank(G) � 2 . Suppose n : [ --> [' is an isomorphism . Then n extends to a rational isomorphism n : G --+ G'
This was first shown for cocompact lattices by [Mostow 2] and then for non cocompact lattices by [Margulis 3].. The theorem is true for IR-rank 1 groups as well excluding PSL(2, IR). This is due to [Mostow 2] for r cocompact, and Mostow's argument was completed to the non-cocompact case by [Prasad 1] However, here we shall only be proving the result under the hypothesis of 5 .. 1 1 Theorem 5 1 1 follows directly from a more general result of [Margulis 1] which deals with the question of extending an arbitrary homomorphism r --> G' to G, without the assumption that the image be a lattice in G' In fact, Margulis's results also give basic information about homomorphisms of r into groups of the form Hk, where k is any local field of characteristic 0 and H is connected, semisimple, and defined over k.. This generalized rigidity result ("superrigidity") is the central part of the proof of Margulis' arithmeticity theorem, which we discuss in the next chapter.
Ergodic theory and semisimple groups
86
5.1.2 Theorem (Margulis' Superrigidity Theorem): Let G be a connected semi simple algebraic IF\-group, IF\-rank( G) � 2, and assume G� has no compact factors Let r c G� be an irreducible lattice. Suppose k is a local field of characteristic 0, and let H be a connected algebraic k-group, almost simple over k. Assume n : r ---> Hk is a homomorphism with n(r) Zariski dense. Then (i) If k IF\, H is IF\-simple (equivalently, center free), and H 11 is not compact, then n extends to a rational homomorphism G ---> H defined over IF\ (and hence defines a homomorphism G11 ---> H11 . ) (ii) If k = C and H is simple (equivalently, center free), then either (a) n(r) is compact (Hausdorff topology); or (b) n extends to a rational homomorphism G ---> H. (iii) If k is totally disconnected, then n(r) is compact (Hausdorff topology). =
For Theorem 5JJ, the relevant case of Theorem 5 . 12 is of course k
=
IF\.
5.1.1 from 5.1.2: Let G be as in 5JJ and G a connected semi simple algebraic IF\-group with trivial center so that G = G� (Proposition 3 . 16). Proof of Theorem
.
We can express each simple factor of G' in a similar fashion, and by applying (i) of 5.. L2 to each simple factor of G', we deduce that n extends to a rational homomorphism G11 -> H11 where H� = G'. (We can apply 5.1.2 since n(r) = r' is Zariski dense in G', and hence in H, by the Bore! density theorem (3..2 ..5)..) By connectedness of G, n(G) c G' . Since n(r), and hence n(G) is Zariski dense in H, and n(G) is an algebraic subgroup of H, it follows that n(G) = H.. For any rational homomorphism over IF\, n( G11) is of finite index in n( G)il'" and so we deduce dim n(G) dim(n(G11))= dim n(G)11 dim H11 = dim G' . Since G'isalso connected, it follows that n(G) = G' . If n is not an isomorphism, let N = ker (n).. Since G is center free, N is of positive dimension, and by irreducibility of r, r is dense when projected to GIN. Since n factors to a map GIN ---> G', this would imply that n(r) is topologically dense in G', which contradicts discreteness of r' c: G'. =
=
.
We begin the proof of Theorem 51.2 for the real field. Proof of 5.1.2 for k
= IF\: We begin by showing that it suffices to find a rational r-map between homogeneous spaces of G and H. (We remark that if H acts on a space, so does r, via the homomorphism n : r ---> H11 . )
5.1.3 Lemma: Suppose P c G and L c H are proper algebraic IF\-subgroups, and that there is a rotational r -map cp : GIP ---> HIL defined over IF\. Then n extends to a rational homomorphism G ---> H defined over IF\
Rigidity
Proof: Let gr(n)
87
=
{(y, n(y) ) E G x H} be the graph of n, and gr(n) the Zariski closure . We claim that gr(n) is the graph of a homomorphism G """""* H We first note that the projection of gr(n) onto G must be all of G, since on one hand this projection contains r and hence is Zariski dense in G by the Bore! density theorem (3.2. 5), and on the other hand is the image under a regular homo morphism of an algebraic group hence must be an algebraic subgroup of G Thus to see that gr(n) is the graph of a homomorphism, we suppose that (g, h l ), (g, h2) E gr(n) .. Let R = R(G/P, H/L) be the space of rational maps G/P ·--> H/L. Then G x H acts on R by [(g, h) cp](x) = cp(xg) · h The condition that cp be a r-map is exactly the assertion that cp is invariant under the action of gr(n). Since the set of points in G x H leaving cp invariant is an algebraic subgroup, this implies that gr(n) leaves cp invariant, and hence, with g, hi as above, cp(xg) = cp(x)h 1 and cp(xg) = cp(x)h 2 . Thus, h 1 h2 1 leaves cp(G/P) pointwise fixed . However, since cp(xy) = cp(x)n(y), n(r) leaves cp(G/P) invariant, and hence it leaves the Zariski closure cp(G/P) invariant Since n(r) is Zariski dense in H, .
H leaves cp(G/P) invariant, and so cp(G/P) must be Zariski dense in H/L Thus, h 1 h2 1 leaves all H/L pointwise fixed . Therefore, h 1 h2 1 E (J hLh - 1 , a normal hEH
subgroup.. Since H is center tree, h 1 h2 1 = e, showing that gr(n) is the graph of a homomorphism. Finally, since the projection gr(n) """""* G is a regular bijective homomorphism and hence has a regular inverse, and the projection gr(n) ·--> H is clearly regular, the extended homomorphism G """""* H is regular. Since n(r) c H n:< and r is Zariski dense in G, n is defined over IR. We are therefore faced with the problem of constructing such a rational map cp We will show that we can do this where P is a minimal parabolic subgroup defined over IR. For example, if G = PSL(n, C), so Gn:< = PSL(n, IR), then P will be the image in PSL(n, C) of the subgroup of upper triangular matrices. In general, we have Gg/Po identified as a Zariski dense subset of G/P where we set Po = Gg n P. Therefore, if we have a rational map cp : G/P ---* H/L which is defined on a Zariski dense subset of Gg/Po and as a map Gg/Po """""* H/L is a r-map, then cp : G/P ---* H/L will also be a r-map. Therefore, it suffices to show that we can find a proper algebraic IR-group L c H and a rational r-map cp : Gg/Po """""* Hn: H is a homomorphism where H is also locally compact, and that there is a measurable r -map ({J : S ---> H/K where K c H is a compact subgroup. Then n(r) is compact
5.1.9
Proposition
be the quasi-invariant probability measure on S, and let v = ({J * (Jl), a probability measure on H/K. The map ([J x qJ : S x S ---> H/K x H/K is a r-map, (({J x ({J) *(/1 x Jl) = v x v, and f1 x f1 is ergodic under r by hypothesis . Thus, v x v is also ergodic under r . We next observe that the H action on H/K x H/K is smooth. To see this, it suffices by , Lemma 3.2 1 3 to see that K is smooth on H/K, which follows from Corollary 2. L2 1 since K is compact Since f acts by n(r) c H, the natural map H/K x H/K ---> (H/K x H/K)/H is r-invariant, and since v x v is ergodic under r and (H/K x H/K)/H is countably separated, this map is essentially constant. In other words, v x v is supported on an H-orbit, say A, in H/K x H/K. By Fubini's theorem, there is a point x E H such that v is supported on { [y] EH/K I ( [x], [ y] ) EA}. But such a set must be a xKx - 1 -orbit in H/K, which is compact. In other words, support(v) c H/K is compact. Since n(r) leaves v quasi-invariant, n(r) leaves support(v) invariant, and since K is compact, n(r) leaves a compact set B c H invariant (under translation).. But then n(r) c BB - 1 , and so n(r) is compact. Proof: Let
f1
To apply this proposition to case (b) in Step 1 ', we need only show that the irreducible lattice r is ergodic on G �/Po x G�/Po . However, the action of Po on G �/ P0 has an orbit of full measure, and with stabilizer A11 11 P 0 , where A is the IR-split abelian subgroup (contained in P) of maximal dimension. (For example, if G = SL(n, IC), and P0 = upper triangular real matrices, let P0 = lower triangular real matrices. Then P0 and P0 are conjugate subgroups of G2 = SL(n, IR), and so we can identify G 2/Po � G2/Po. Our assertion then follows from the discussion following lemma 5.1.4.) Thus, the G 2 action on G2/P0 x G �/P0 has an orbit of full measure and with stabilizer A11 11 P0 .. Thus, as a measurable r-space, G2/P0 x G�/Po � G g/A 11 11 P0. However, since A11 11 Po is not compact (for SL(n, IC), this is just the real diagonal subgroup), r is ergodic on this space by Moore's theorem (2.2. 7). Thus, to complete the proof of Theorem 5.1 2 for k #- IR, it remains only to consider case (a) in Step 1'. For k = C, exactly the same argument as the proof for k = IR now goes through and we thus deduce that n extends to a rational homomorphism G -+ H. For k #- IR, C, i.e , k totally disconnected, the proof of Step 2 goes through if we replace "rational" by "constant". The point is that since each ( U;)11 is connected, any homomorphism of ( U;)11 into the k-points
Rigidity
95
of a k-group (which is totally disconnected) must be constant Therefore, in the case k ¥- IR, C, we deduce the existence of an essentially constant r-map q; : G 2/P0 ---+ Hk/Lk, where L e H is a proper k-subgroup. This means that there is a fixed point for n(r) acting on Hk/ Lk, i e , n(r) is contained in a conjugate of Lk But this contradicts the assumption that n(r) is Zariski dense in Hk Hence, for k ¥- IR, C, the only case which can occur in Step 1 ' is case (b), and hence n(r) is always compact This completes the proof of Theorem 5. 1 . 2 in all cases. Remarks: Margulis' first proof of Theorem 5J.. 2 appeared in [Margulis ll The
proof of Step 2 that we have presented is essentially the proof of that paper. Margulis' original proof of Step 1 was based on the "multiplicative ergodic theorem" of [Oseledec l J (See also [Raghunathan 2].) In [Margulis 2], a different approach to the proof of Step 1 was presented, using the existence of measurable r-maps G 2/ P0 ---+ M(X ) where X is a compact metric r-space Here, we have seen this property as a consequence of amenability of r acting on G2/P0 ; this particular consequence of amenability for this action was first established in [Furstenberg 2]. The proof of Step 1 we have presented here based upon amenability and the smoothness of actions of algebraic groups acting on suitable spaces of measures, is from [Zimmer 8]. This latter proof was developed in order to be able to extend the super-rigidity theorem to cocycles of ergodic actions. (Cf.. the remarks following the proof of Proposition 4..21 6.)
5.2
Rigidity and orbit equivalence of ergodic actions
In this section we shall prove the following rigidity theorem for ergodic actions. Theorem [Zimmer 8]: (Rigidity for ergodic actions of semisimple Lie groups). Suppose G, G' are connected semisimple Lie groups with finite center and no compact factors. Suppose S (resp. S') is an essentially free ergodic irreducible G (resp. G')-space with finite invariant measure, and assume that the actions are orbit equivalent . Assume IR-rank(G) � 2. Then
5.2.1
(i) G and G' are locally isomorphic.. (ii) In the centerjree case, G � G', and identifying G and G' via this isomorphism, the actions of G on S and S' are isomorphic. Remarks: (a) We saw in section 4.2 (see the paragraph following the proof of
Proposition 4.216) that the problem of extending homomorphisms defined on
96
Ergodic theory and semisimple groups
subgroups and the problem of showing that orbit equivalent actions are iso morphic are closely related, and in fact are both special cases of a problem concerning cocycles. We shall deduce Theorem 5.2. 1 from a general super rigidity theorem for cocycles, which will then subsume both Theorem 5.2.. 1 , and the Mostow�Margulis Theorem (5. 1 1 ), and generalize Margulis' super rigidity Theorem (5.. 1.2).. We indicated in section 51 that Margulis' superrigidity theorem can be applied to problems other than the classical rigidity problem in 5. 1 . 1 , e. g. to arithmeticity which will be discussed in Chapter 6. Similarly, the superrigidity theorem for cocycles we will prove (Theorem 5.2 .5 below) can be applied to problems other than orbit equivalence.. See for example, section 94, and in a more geometric direction [Zimmer 1 5, 1 8, 1 9] (b) At the conclusion of section 4. 3, we described the orbit equivalence theory for amenable actions. The salient feature of that theory is that orbit equivalence is a very weak condition for amenable actions. In particular, for actions of discrete amenable groups with finite invariant measure, orbit equivalence of actions in no way enables one to distinguish the acting groups, and even given the group, in no way enables one to distinguish the actions. Thus, Theorem 5.2J shows that the orbit equivalence theory for semisimple Lie groups is in a sense diametrically opposed to the theory in the amenable case (c) The rigidity oflattices in semisimple groups can be considered as a generaliza tion of a type of rigidity of lattices in Euclidean spaces . (This is the point of view in the introduction.) On the other hand, the rigidity of ergodic actions of semi simple groups is not a generalization of a phenomenon for actions of Euclidean groups, as our discussion in (b) indicates . (d) We indicated in section 5J that the rigidity theorem for lattices ( 5J J) is also true for simple groups of IR-rank 1 as long as we exclude PSL(2, IR). It is there fore natural to enquire as to whether or not Theorem 5.2 . 1 is also true for these groups. While some partial information has been obtained ( [Zimmer 1 6] ), this question is open as of this writing. Before turning to the proof of 5 . 2. 1 , let us present an example.
5.2.2 Corollary [Zimmer 8]: For i = 1 , 2, let Gi be a connected simple Lie group with finite center, r; c Gi a lattice, and Si an essentiallyfree ergodic C-space with finite invariant measure. Suppose IR-rank(G 2 ) � 2, and that the r 1 action on S 1 and the r 2 action on Sz are orbit equivalent. Then G 1 and G 2 are locally isomorphic.
Proof: Let X i be the G i action induced from the C-action on S i (Definition 4.2..21). Since r; c G i is a lattice, and ri acts with finite invariant measure on Si , Gi acts
97
Rigidity
with finite invariant measure on Xi It is a straightforward consequence of the definition that Gi will be essentially free on Xi if [; is so on Si. It is also straightforward to see that if x i = Gi/C X ,,si (as in Definition 4.2. 21), then x, yE X i are in the same Gi orbit if and only if their projections to Si are in the same [ ; orbit Thus, if 8 : S 1 --+ Sz is an orbit equivalence, and f: G I /f 1 --+ G 2 jf2 is a measure class preserving Bore! isomorphism (Appendix A), then the map
Hk is a cocycle such that a is not equivalent to a cocycle taking values in a subgroup Lk where L c H is a proper algebraic k-subgroup.. Then
(i) If k = IR, H is IR-simple (equivalently, centerfree), and HrK is not compact, then there is a rational homomorphism n : G --> H defined over IR such that a � a " I G�, i e , a is equivalent to the cocycle (s, g) --> n(g). (ii) If k = C and H is simple (equivalently, center free), then either (a) a is equivalent to a cocycle taking values in a compact subgroup of H; or (b) there is a rational homomorphism n : G --> H such that a � a" I G� · (iii) If k is totally disconnected, then a is equivalent to a cocycle taking values in a compact subgroup of Hk As in the situation of lattices, the relevant case for the rigidity theorem for actions (5..2. 1 ) is the case k = R Before we deduce 5..2 . 1 from 5..2..5, let us first observe that Margulis superrigidity (5 . 12) also follows from 5.2.5. To see this, suppose n : r --> Hk is a homomorphism as in 5 . 1.2, and let a : G2/r x G2 --> Hk be a strict cocycle corresponding to n (Proposition 4. 2 . 1 3).. We can then obtain 51 ..2 from 5..2..5 by applying the latter and Proposition 4..2. 1 6 to a once we observe the following general fact .
.
.
Suppose G is locally compact, G0 c G a closed subgroup, and G --> H a cocycle corresponding to a homomorphism n : G0 -+ H (4.2.1 3, 4. 2 . 1 5). If L c H is a subgroup, then a is equivalent to a cocycle taking values in L if and only if n(G 0) is contained in a conjugate of L. 5.2.6
Lemma:
a : GjG0
x
Proof: This follows from 4.. 213, 42. 1 5
99
Rigidity
Proof of Theorem 5.2.1 from 5.2.5: Let G, G' be as in Theorem 5.2. 1 . We first
consider the case in which both groups have trivial center. We wish to apply Proposition 4. 2J 1 We can write G � G � where G is a connected semisimple algebraic IP-group with trivial center.. Let H be a simple factor of G', and write H = Ij �, where Ij is a connected simple adjoint IP-group. Let rx : S x G -. G' be the cocycle corresponding to an orbit equivalence 8 : S --+ S' (Example 4. 2..8).. We first claim that rx is not equivalent to a cocycle into a proper algebraic subgroup of G' . If it were, we could write rx(s, g) = cp(s) - 1 fJ(s, g)cp(sg) where cp : S --+ G' and fJ(s, g) E L', L' c G' proper algebraic, and for each g the identity holds a.e. Define a map A : S' -. G' by A(y) = cp(8 - 1 (y)).. The map (s, g) -. (8(s), rx(s, g)) is a measure class preserving bijection between conull subsets of S x G and S' x G'. (This would be clear if G, G' were discrete. For the general case, see [Zimmer 9] ).. Since cp(s)rx(s, g)cp(sg) - 1 E L' for almost all (s, g), we have A(y)g'A(yg') - 1 E L' for almost all ( y, g') ES' x G' . In other words, A(y)g' = A(yg') in G'/L' for almost all y, g' . If we let f1 be the G'-invariant probability measure on S', then viewing A as a map S' --+ G'/L', we then have that for almost all g', A * (/1) is g'-invariant Since a conull subgroup of G' must be all of G' (Appendix B) this implies that A * (/1) is a G'-invariant probability measure on G '/L' If L' is proper, this is impossible by the Borel density theorem (32 . .5) . This verifies our first assertion about rx. We also observe that if A is any automorphism of G ', then A o rx is not equivalent to a cocycle into a proper algebraic subgroup. This actually follows from the above paragraph since A o rx is the cocycle corresponding to the orbit equivalence 8 : S -. S', where G' now acts on S' by (y, g') -. yA(g'). Now let p : G' --+ H be the projection and consider the cocycle p o rx : S x G --+ H. If this were equivalent to a cocycle into a proper algebraic subgroup L c H, then rx would clearly be equivalent to a cocycle into p - 1 (L). (Just write G' = H x H' and examine the definitions . ) Similarly, if A is any automorphism of H, then A o p o rx:S x G --+ H is not equivalent to a cocycle into a proper algebraic subgroup (since A defines an automorphism of G' by taking the identity automorphism on H').. We wish to apply Theorem 5 . .2..5 to the cocycle f3 = p o rx (for each simple factor of G'). The hypotheses and conclusions of 5.2.5 concern cocycles into Ijr?., and here we have cocycles into H Ij�. To compare them, we use the following. =
5.2.7 Lemma: Suppose G, H, J are locally compact groups with J c H normal of finite index. Suppose rx:S x G --+ J is a cocycle on an ergodic G-space. Then (i) If rx:S x G --+ H is equivalent (as a cocycle into H) to a cocycle taking values in a subgroup H 0 c H, then there is an element h E H such that, letting Ah be conjugation
1 00
Ergodic theory and semisimple groups
by h, Ah o r:t. is equivalent as a cocycle into J, to a cocycle taking values in J n H 0 (ii) If r:t., f3:S x G � J are cocycles which are equivalent as cocycles into H, then for some h E H, Ah o r:t. and f3 are equivalent as cocycles into J. Let us postpone the proof of this lemma for a moment and continue our previous argument From (i) of the lemma, and our previous observations, it follows that as a cocycle into lj ""' p o r:t. cannot be equivalent to a cocycle into a proper algebraic subgroup LIP. c lj lP. For if it were, by (i) of the Lemma, LIP. ::::J lj�, so dim L = dim lj, and L = lj by connectedness of lj. We can therefore apply the case k = IR of Theorem 5.2..5, and deduce that p o et. "' r:t."IG as cocycles into ljiP. for some IR-rational homomorphism n:Q � lj. We observe that since G is connected, n (G) c H = lj� Thus by (ii) of Lemma 5..2.7, replacing n by Ah o n for some h E H, we have an IR-rational homomorphism such that p r:t. "' r:t."1 a as cocycles into H = lj� Since this is true for each simple factor of G', there is a rational Qomomorphism it:G ·--> G' such that r:t. "' r:t.n as cocycles into G' . By Proposition 4. 2. 1 1, to complete the argument (in the center free case), it suffices to show that it is an isomorphism. Since r:t. is not equivalent to a cocycle into a proper algebraic subgroup, if(G) cannot be contained in a proper algebraic subgroup . Arguing as in the conclusion of the proof that Theorem 5 . 1 . 2 implies 5 .. U in the preceding section, we deduce that it (G) = G' . Finally, suppose N = ker (if).. From r:t. "' r:t.n, we deduce that there is a function H is projection, we obtain as in the center free case a rational homomorphism ii:G/Z(G) --> G '/Z (G') such that 11 � an It follows that [3, and hence a, is equivalent to a, where n:G --> G '/Z(G') is the composition of ii with the natural projection . One sees n is surjective exactly as in the center free case, and to show that N = ker (n) is actually Z (G), we use the same argument as in the center free case applied to the equation
il(s) a (s, g) = lJ(sg) where lJ:S --> S'/Z (G') is the composition of e with the natural projection. If N -:f. Z(G), then the argument would show that G '/Z(G') is essentially transitive on S'/Z(G') which would again imply essential transitivity of G' on S' providing a contradiction. Thus, to complete the proof of Theorem 5.21 from 5 ..2. 5, it remains only to verify the lemma
a :S x G --> H is equivalent to a cocycle taking values in 1 and equivalent to a cocycle taking values in H0 By Example 4..2J 8, there are a-invariant functions f:S --> H/H0 and f':S --> H/1. Then cp = (f, f') : S -> H/Ho x H/1 is also a-invariant Thus (taking H to act on the left on the coset spaces), for all g, a (s, g) cp (sg) = cp (s) a . e. This implies that cp (sg) = cp (s), where cp:S -> (H/Ho x H/1)/H. The action of H on H/H0 x H/1 is smooth (since by 3..213, it suffices to see that Ho is smooth on H/1, which it dearly must be since H/1 is finite), i.e. , (H/H0 x H/1)/H is countably separated . By the ergodicity of G on S,
H/1 n H 0 By Example 4..2J 8, a is equivalent (as a cocycle into H) to a cocycle f3 taking values in 1 n H0. Thus, to complete the proof of assertion (i) of Lemma 5 2. 7, it suffices to prove assertion (ii).. Choose a function cp:S -. H such that for each g, f3 (s, g) - 1 cp (s)a (s, g) = cp (sg) a. e. Thus, qi (s) = qi (sg) where qi is the composition of cp with the projection H --> 1\H/1. Since 1\H/1 is finite and G is ergodic on S, qi is constant on a Proof of Lemma 5.2.7: The cocycle
102
Ergodic theory and semisimple groups
conull set, say
Hk/M such that for all g, cp (sg, xg) = cp (s, x) a(s, g) a.e , where either (a) M = Lk where L c H is an algebraic k-group of strictly lower dimension; or (b) M is compact.
107
Rigidity
As in 5. 1 .2, if k = C, and we are in case (a) in Step 1 ', the proof for k = lR goes through. If k f= lR, C, and we are in case (a), then as in 5J .2, for almost all s, the map cps:G�/Po -" Hk/M is constant Thus, in this situation, we can consider cp as a map cp:S -" Hk/M such that for all g, cp (sg) = cp (s) cx (s, g), i e , cp is an et-invariant function (after we switch to a left action of H on Hk/M). (Definition 4.21 7. ) By Example 4 2J 8(b), this implies that et is equivalent to a cocycle into M = Lk, which is impossible given our hypothesis about et . Thus, case (a) in Step 1' is impossible for k =!= lR, C. It remains only to consider case (b), and for this we use the following generalization of Proposition 5J.9. .
Let G be locally compact, S, Y ergodic G-spaces such that G still acts ergodically on S x Y x Y Let ct:S x G __,. H be a cocycle, where H is locally compact, and suppose there an &-invariant function cp:S x Y __,. H/K where K c H is compact and fi.(s, y, g) = ct (s, g). Then et is equivalent to a cocycle into a compact subgroup of H 5.2.10
Lemma:
To see that this suffices to complete the proof in case (b) above, it suffices to check that G� acts ergodically on S x G�/Po x G�/P0 , where S is an irreducible ergodic G �-space. However, as we observed following the proof of Proposition 5. L9, as a measurable G �-space G �/Po x G�/Po is isomorphic to G �/A rr1 n Po Thus, by Proposition 2.22, it suffices to see that Arr1 n Po is ergodic on S, which in turn follows from the noncompactness of Arr1 n P0 and Moo re's theorem 2.2. 1 5. We now turn to the proof of 5..2.10. Proof of Lemma 5.2.10: Let
m be a quasi-invariant ergodic measure on S, and
J1
a quasi-invariant ergodic measure on Y We let H act on the right on H/K, so that &-invariance of cp is the assertion that for each g and almost all (s, y), cp (sg, yg) = cp (s, y) et (s, g).. Consider the map ip:S x Y x Y __,. H/K x H/K, ip (s, Yl , Yz ) = (cp (s, y l ), cp (s, Yz )). Let p:H/K x H/K -" (H/K x H/K)/H be the natural map. We recall (see the proof of Proposition 5. 1 .9) that (H/K x H/K)/H is countably separated since K is compact The a-invariance of cp implies that p o ip is essentially G-invariant By the assumption that G acts ergodically on S x Y x Y, p o ip is constant on a conull set, i e , there is a single H-orbit in H/K x H/K such that ip (s, y 1 , Yz ) lies in this orbit for almost all (s, y 1 , y2 ). For each s E S, let i/Js: Y x Y--" H/K x H/K be given by ip,(yl, Yz) = ip (s, y 1 , Yz). Define cp,: Y __,. HIK similarly . Then ip, = ( cp,, cp,) . By Fubini's theorem, for almost all s, ip s (y 1 , Y z) lies in the distinguished H-orbit in H/K x H/K for almost all y 1 , Yz In other words, (ip,) * (Jl x Jl) = (cp,) * Jl x (cp,) * Jl is a measure supported on this orbit Arguing as in the proof of Proposition 5. L9, we deduce that for almost all s, the support of (cp,) * (Jl) is a compact subset of H/K Let rtJ be the space of .
Ergodic theory and semisimple groups
108
compact subsets of H/K with the Hausdorff metric . (See e.g. 3:2. 6.) Then Cfl is a separable metrizable H -space . The map s --> A, = support ( ( Cfl, and since
B, then since A, B, K are compact it follows that { hn} lies in a compact subset of H. Thus, we can assume hn --> h E H, and so B = A h. Thus, all H-orbits on Cfl are closed, verifying smoothness. Since :S -> rl is a-invariant, i e , for all g, (s) o:(s, g) = (sg) a. e. (using a right action), :S --> Cfl/H is essentially G-invariant By ergodicity of G on S, and the fact that Cfl/H is countably separated, is essentially constant, i e , there is a single H-orbit in Cfl such that for almost all s, (s) lies in this H-orbit. Thus, we can consider as a a-invariant map :S --> HIM where M is the stabilizer of an element in Cfl lt is clear that M must be compact since K is compact It then follows from Example 4..2.1 8(b) that a is equivalent to a cocycle into M . This completes the proof of Lemma 5 2.10, and hence, completes the proof of Theorem 5.2.5 in all cases.
Remark: The reader will observe throughout the proof of Theorems 5 . L2 and 5.2..5 the constant interplay between ergodicity of certain actions and
smoothness of other actions. One way in which this was used was to show that a cocycle was equivalent to one that took values in a subgroup. This type of result is generally useful, and we can formalize the technique we have employed in the following..
(Cocycle Reduction Lemma). Suppose a:S x G -> H is a cocycle into a locally compact group H where G acts ergodically on S. Suppose X is a continuous H-space on which H acts smoothly. lf there is an a-invariantfunction X, then there is a point x E X, (with stabilizer Hx) such that a is equivalent to a cocycle taking values in Hx. 5.2. 1 1
Lemma:
Proof: We have for each g,
(eN)a given by p = T- 1 (S(p" 1 o q 1 , . . p"d o q ) ), for p E k[X 1 , . ' X ] n l.. The point of this a N construction is that p o T is actually a polynomial (into (eN)a) with coefficients in ro . To see this, it suffices to see that p(T((rpN )d)) c (rp N)d, i.e. p((kN)') c (rp N )d But if x E k N, we have .
X
o
'
We have S(a 1 (p(x)), . , ad(p(x)))E(k N )', and hence upon applying T- 1 we obtain an element of (rp N)d Therefore, under the identification of (eN)a with (eN)a via T, this variety ll V"i is actually defined over rp. Furthermore, (ll V"i)(Q = ( Vk)' Where Vk iS the image Of Vk under the isomorphism kN -> (k N ) We summarize our discussion in the following. (We shall only be using this construction for groups, and hence just summarize for this case. ) '..
6.1.3 Proposition (Restriction of Scalars) [ Weil l] Let k be an algebraic number field ( [k:rp] = d), and suppose G is an algebraic k-group.. Let a 1 , . . . , e be
the distinct field embeddings of k into e with a 1
d
identity. Let Rkt(Q(G) = fl G"', i= 1 and for g E Gk let g' = (a 1 (g), . , aa(g)). Let (Gk)' = {g'l g E Gk } . Then Rkt(Q (G) is (isomorphic to) an algebraic rp-group such that (Rkt(Q (G))(Q = (Gk)', and (Rk;(Q (G))z = (GI'!)' where (!) c k is the subring of algebraic integers in k. The projection map p:Rkt(Q(G) -> G onto the .first factor is defined over k, and defines bijections (Rk/(Q (G))(Q -> Gk, (Rk/(Q(G))z -> GI'! Furthermore, if for all i, a i (k) c K where K is a field k c K c e, then each Ga, is defined over K and (Rk/(Q(G))K = ll (Ga'k The group Rk/(Q(G) is called the restriction of G to rp. Theorem 3.1.7 implies: =
1 17
Margulis' arithmeticity theorems
Corollary: If k is an algebraic number field, and G is a semisimple algebraic k-group, then Gl!l is isomorphic to a lattice in [Rk; (O"(a;i) ), and since G and L(G) are defined over H;. In the for mer case, all eigenvalues of Adn,(p;(O" (y) ) ) ) have absolute value 1, and hence I Tr (Adn, (p;(O" (y) ) ) )l � dim H;. If p; o O" extends to n, then letting Cg : G -. G and ch:H; -> H; be conjugation by g E G and h E H;, n o Cg = c"(gJ a n, and hence upon differentiating, we obtain for each g E G the commutative diagram L(G) dn
Ad(g)
1
1
L (H;)
L(G)
Adn.(n(g))
dn
L(H;)
Since dn is surjective, any eigenvalue of Adn, (n(g)) is an eigenvalue of Ad (g), and in particular, for y E 1, any eigenvalue of Adn, (p;(O"(y) ) ) is an eigenvalue of A d (y).. Thus, if we let e (y) be the maximal absolute value of an eigenvalue of Ad (y), we deduce that for any O" E Aut (C), I Tr (Ad(O" (y) ) ) l � (1 + e (y) ) dim G. This proves the lemma The next step is to pass from the conclusion that the traces are algebraic to the assertion that we can assume that the matrix entries themselves are algebr aic.
Let K be the field of real algebraic numbers. Then for some m, there exists a faithful IR-rational representation n:G -> GL(m, q such that 6.1 .7
Lemma:
Margulis' arithmeticity theorems
1 19
n(r) c GL(m, K ). Thus, identifying G with n(G), we can assume that G is defined over K and r c G K Proof: Let T:G ---> C be given by T(g) = Tr (Ad (g) ). This is a polynomial function, and hence the linear span V of { g Tlg E G} is finite dimensional, where g acts by translation on polynomial functions, i e , (g T)(h) = T(hg). Let n be this representation of G on V Assume for the moment that n is faithful. Since r is Zariski dense in G, V is also spanned by { n(y) Tly E r}, (for otherwise we would have a r-invariant subspace under a rational representation of G which was not G-invariant) Choose y 1 , , Ym E r such that { n (y;) T} is a basis for V It suffices to show that with respect to this basis the matrices of n (y) have all entries in K, for all y E r. Since elements of V are polynomial functions, and { n(y;) T} are linearly independent as functions on G, the Zariski density of r in G implies that they are linearly independent as functions on r. Therefore, we can find s 1 , . , Sm E r such that the matrix A = (n(y;) T)(si)) = (T(sm)) is non singular.. By Lemma 6.1 . 6, A E M (m x m, K) . Let C (y) = (c;i(Y)) be the matrix of n (y) with respect to { n(y;) T} . Thus, n (y)(n (y;) T) = L cii(y)(n (yi) T).. Evaluating
j L c;i(Y) T(skyi). Lemma 6.. 1.6 implies j that T(skyy;) E K, so for some matrix B E M (m x m, K), we have B = C(y) A. Since A E M (m x m, K) as well, and A is invertible, we deduce C(y) has entries this equation at sk, we obtain T(skyy;)
=
in K. Thus, to verify lemma 6.. 17 it suffices to show that the representation n is faithfuL Suppose g E ker(n).. Then g T = T, i.e. Tr(Ad(hg)) = Tr(Ad(h)) for all h E G. Let W be the linear span of {Ad(h) I h E G } in the space of all endo morphisms of the linear space L(G). We then have Tr(MAd(g)) = Tr(M) for all M E W Since G is semisimple, Ad is a direct sum of irreducible representations. Thus we can write the representation (Ad, L(G)) � LEB(O";j, V;i) where each O";j is irreducible, (O"; j , V;i) = (O";k, V;k), and for i i= r, O"ik and O",k are inequivalent Any endomorphism of LEil V;i leaving each V;i invariant is given by a family M;i E End( V;i). From the irreducibility of O";i it follows (by Wedderburn's theorem) that under the above identification L(G) � LEB V;h we have
W = { M E End(LEil V;i) I M( V;i) c V;i all i, j, and M;i = M;k for all i, j, k } . From the fact that Tr(MAd(g)) = Tr(M) for all M E W, we then deduce that Tr(SO";/g)) = Tr(S) for all S E End( V;i)· It follows that O";/g) = I for all i, j, and hence that Ad(g) = I . Since G has trivial center, g = e completing the proof We will now use the fact that the group r is finitely generated . We will prove
120
Ergodic theory and semisimple groups
this in the case in which all simple factors Of G have IR-rank at least two in chapter 7 . (See 7. 1.5 . )
6.1.8 Lemma: With G as in the conclusion of 6. 1 . 7, there is a real algebraic number field k ( [k Q] < oo) such that G is de,fined over k and f c Gk Proof: If { yi} is a finite set of generators of f, let k be the field generated by the matrix entries of Y i Then by 61 7, [k : Q] < oo, and clearly r c GL(m, k). Since f is Zariski dense in G, it follows from Proposition 3. L8 that G is defined over k
We now complete the proof of the theorem. Let Rk;CQ(G)
=
flG""; be as in i
Proposition 6. 1 3.. Let a : Gk --> Rk;CQ(G) be the map a(g) = ( G, and a homomorphism a : r --> HCQ such that p o a is the identity and a(f) is Zariski dense in H Since G is connected, p(H 0) = G, and hence by replacing r by the finite index subgroup f n a - 1 (H 0), we can also assume that H is connected. We now claim that (ker p)fR is compact We have an IR-isomorphism of algebriac IR-groups H � G x (ker p) in such a way that p corresponds to projection onto G. Let F be an IR-simple factor of ker p, so we can write H � G x F x F' as algebraic IR-groups, where F' is the product of the remaining IR-simple factors of ker p . Let q : H --> F be projection. Since a(f) is Zariski dense in H, (q o a)(f) is Zariski dense in F, and since q is an IR-map, (q o a)(f) c FrR·· We claim FfR is compact. If not, then by the case k = IR of Margulis' superrigidity theorem (5. 1 .2), q a a extends to a rational homomorphism h : G --> F. But then { (g, h(g), f') I g E G, f' E F ' } would be a proper algebraic subgroup of H containing a(f), contradicting Zariski density of a(f) in H. This shows that F fR is compact for any IR-simple factor F of ker p, and hence (ker p)fR is also compact For any prime number a, let Oa be the field of a-adic number s. We have a natural embedding HCQ --> H G can in fact be taken to have not only compact kernel, but trivial kernel, i.e. , p can be chosen to be an isomorphism. The proof of this from Theorem 6.. 17 depends on the following compactness criterion, which should be considered as a companion result to Theorem 31 ..7. 6.1.9 Theorem [Borel�Harish�Chandra 1 ] , [Mostow�Tamagawa 1 ] : Let G be a semisimple algebraic (Q-group. Then GR/Gz (which is of finite volume by 3 . 1.7) is compact if and only if the only unipotent element in Gro is the identity.
We then deduce Corollary [Margulis 1]: If G is a connected semisimple algebraic IR-group with trivial center and IR-rank(G) � 2, and r c cg is an irreducible non cocompact lattice, then there is a connected semisimple algebraic (Q-group H with trivial center and an isomorphism p · HR --> GR such that p(Hz) and r are commensurable. 6.1.10
Proof: By Theorem 6. L2, we can find a connected semisimple algebraic
(Q-group H with trivial center and an IR-rational surjective homomorphism p : H --> G such that p(Hz) and r are commensurable and (ker p)R is compact. If r is not cocompact in cg, then Hz is not cocompact in HR Thus, by
Ergodic theory and semisimple groups
122
Theorem 6.. 19, H'f:i possesses non-trivial unipotent elements. Since ker p is a normal IR-subgroup, there is a normal IR-subgroup L c H such that the product map L x ker p � H is an isomorphism of IR-groups. If we let q : H � ker p be the projection map then q is IR-rational, and hence if x E HIf:! is unipotent, so is q(x) E (ker p)11. Since (ker p)111 is compact, q(x) = e, and hence H'f:i n L i= {e} Since L c H is normal, H IQ n L is normal in HIf:! , and since HIQ c H is Zariski dense (by 3 1 . 9) the Zariski closure of H IQ n L, say L 1 . is a normal subgroup of H. Since L 1 contains (by definition) a Zariski dense set of �-points, L 1 is a �-group by Proposition 1 L8.. Thus, (L 1 )z is a lattice in (L 1 )111 by 31 .7, and since p : L � G is an isomorphism, p( (Ll )z) is a lattice in p(L 1 )111 and in particular is infinite. Since p(Hz) and r are commensurable, this implies r n p(L 1 )111 is infinite. Because r is an irreducible lattice in Gg, this implies that p(Ll) = G. (This follows from the general fact that if G
=
n
fl G i is a product of connected simple
i= 1 Lie groups with trivial center and r c G is an irreducible lattice, then m
r n TI G i = { e} as long as m < n. This is so because r0 is normalized i= 1 by r and TI Gi, and hence r 0 is normal in G since r TI G i , is dense r0
=
i>m
i>m
b y the irreducibility assumption. Since G i are center free and r 0 i s discrete, this is impossible unless r0 is triviaL) Thus, L1 = L, and hence L is actually defined over � · Thus Lz is a lattice in L111 , and since (ker p)111 is compact, Lz is a lattice in H 111 Thus Lz c Hz is a subgroup of finite index, and p(Lz) and r are commensur able. Thus, replacing H by L, we obtain the required �-group.
6.2
The commensurability criterion
In this section we present a result of Margulis which gives a necessary and sufficient condition for a lattice to be arithmetic without the assumption that the IR-rank of the ambient Lie group is at least two .
Definition: Let G be a locally compact group and r c G a closed subgroup. Let CommG(r) = {g E G i grg - 1 and r are commensurable } . Then CommG(r) is called the commensurability subgroup of r in G If G is understood, we shall sometimes denote this simply by Comm(r) 6.2.1
..
We remark that Comm(r) is a subgroup of G and r c Comm(r) c G. If r, r' c G with r and r' commensurable, then Comm(r) = Comm(r'). As an example, we have the following.
Margulis' arithmeticity theorems
123
[Borel 5]: Let G be a connected semisimple algebraic �-group with trivial center. Then G� c Comm(Gz) and if G" has no compact factors, then G� = Comm(Gz).
6.2.2
Proposition
Proof: Let g E G�, and let m be the least common multiple of all denominators of the entries of the matrices g, g - l Let r = { y E Gz I y = I mod m 2 }, so that r c Gz is clearly of finite index. If y E r, then y = I + m 2 B where B is an integral matrix, and hence gyg - 1 = I + m 2 gBg- 1 which is also an integral matrix . Thus grg - 1 c Gz Furthermore, since r is of finite index in Gz, r is a lattice in G" and hence so is g rg 1 . Therefore, g rg - 1 must be of finite index in Gz Since g rg - 1 is clearly of finite index in gGzg- 1 , it follows that g E Comm( Gz) To see the converse, let C [G] and �[G] denote as usual the space of regular functions on G and the space of �-regular functions on G respectively.. Let V be the subspace of C[G] spanned by the matrix coefficient functions, i e , g ---> g;h and V� the subspace of �[G] spanned by these functions. Then for some m, we can identify V � c m in such a way that V� � � m Define a representation of G on C[G] by (n(g)f)(x) = f(g - 1 xg) . Then Vis G-invariant, and n : G ---> GL( V) is a rational representation defined over �. Since G has trivial center, n is faithful, and thus n is a �-isomorphism of G with n(G). Thus, if g E G, to see that g E G� it suffices to see that n(g)fE V� for all fE V� Suppose that g E Comm(Gz) and [E V� Then we can write n( g)f = -
.
.
n
fo + L: c;f;, where fi E V�, c; E C and (1, c 1 , . . , c.) are linearly independent over i= 1 �·· Since g E Comm(Gz), gGzg- 1 n Gz is a lattice in Gp, and hence by the Borel density theorem (.1 2.5), gGzg - l n Gz is Zariski dense in G. If x E g Gzg - l n Gz, then f(g - 1 xg) = fo(x) + L;cdi(x) where f(g - 1 xg), fo(x) E � By linear independ ence of { 1 , c;} over �, we deduce that fi(x) = 0, f(g - 1 xg) = f0(x) for all x in a Zariski dense set Thus, this holds for all x E G, and hence n(g)f E V� By the conclusion of the preceding paragraph, g E G�, and this completes the proof. .
For an irreducible lattice, we have the following dichotomy.
Let G be a connected semisimple Lie group with trivial center and no compact factors, and let r c G be an irreducible lattice.. Then either r c:: Comm(r) is a subgroup of finite index or Comm(r) is dense in G (with the Hausdorff topology). 6.2.3
Proposition:
= Comm(r) and H 0 the (topological) connected component of the identity. Since r c H, r normalizes H 0 , and hence L(H 0 ), the Lie algebra of Proof: Let H
124
Ergodic theory and semisimple groups
H 0 , is an invariant subspace of L(G) under Ad(r). Since r is Zariski dense in G, L(H 0) is invariant under Ad(G), and hence G normalizes H 0 If H 0 = {e} , then H and hence Comm(r) are discrete, and since r c Comm(r) c G and r is a lattice in G, [Comm(r): r] < oo . On the other hand, if H 0 is a non-trivial normal subgroup then rH 0 is dense in G by irreducibility of r, and since rH 0 c H, G = H, completing the proof For G = SL(n, IR), and r = SL(n, Z), it is clear from Proposition 6.2 2 that Comm(r) is dense in G with the Hausdorff topology. The next proposition asserts that this is true in a more general setting Proposition: Suppose G, r are as in Proposition 6. . 2 . 3 . If r is arithmetic, then Comm(r) is dense in G .
6.2.4
Proof: Let H be a connected semisimple CQ-group and p : H g --+ G a surjective homomorphism with compact kernel such that p(Hz n H 2 ) is commensurable with r. By 6. 2.3 it suffices to show that Comm(r) is not discrete. It suffices to see that p(CommHrFlHz)) is not discrete, and hence by 6.2. 2 that p(Hrq) is not discrete. To establish this, we call on a result of [Bore! 6] described in Chapter 10.. Namely, if a E Z is a prime number, let Z(a) c CQ be the subring consisting of all rational numbers whose denominator is a power of a. We have natural non discrete embeddings of Z(a) into IR and COa, where COa is the field of a-adic numbers. However, the image of Z(a) in IR x COa will be discrete and in fact will be a lattice in IR x COa · [Bore! 6] establishes that the analogous result is true for H. Namely, under the natural embedding A : Hz sup J (Re f)cp = ess sup (Re( f ) ). But this contra-
diets (b) in Definition 7..2. 1.
0, there is t/J E P( G ) c L 1 (G) such that 1 1 n(g)t/1 - t/J II 1 < B Let cp = t/1 1 12 . Then cp is a unit vector in U(G), and J i cp(xg) - cp(xW ;;:; fl t/J(xh) - t/J(x)l .. (This last asser tion just follows from the fact that for a, b � 0, la - b l 2 ;;:; la2 - b21 ) Thus 1 1 n(g) cp - cp l l z ;;:; £ 1 12 From this and the preceding paragraph it follows that the regular representation almost has invariant vectors . This completes the proof of 7 . L8 for G discrete We now turn to the proof of 7.. 1 8 for general locally compact (separable) G. If we examine the proof for G discrete, we see two problems in extending it to the continuous case. The first problem is that it is not immediately clear that there is a G-invariant mean on L 00(G), because the action of G on L 00(G) is not continuous (in G), and hence the argument in the discrete case will not apply directly. The second problem is that to establish that the regular representation almost has invariant vectors, we need to control 1 n(g)cp - cp ll z uniformly for g in a compact set In the discrete case, it therefore sufficed to do this pointwise in G.. Thus, this part of the argument also requires modification. It will be convenient now to switch to the left action of G on itself. Thus, regular representation will now mean left regular representation (i.e., (n(g)f)(x) = f(g - 1 x)) with respect to left Haar measure, and similarly the action on spaces of means will be derived from the left translation on spaces of functions. We recall that f: G -> C is called (left) uniformly continuous if for all B > 0, there is a neighborhood U of e E G such that lf(yx) - f(x) l < �> for all x E G, y E U. Let UCB(G) denote the space of (left) uniformly continuous bounded functions on G.. Then we have UCB(G) c L00(G ), and the action of G on UCB(G ) is continuous . UCB(G) is not separable, but arguing as in the first paragraph of the proof of 7. 1 . 8 for G discrete we see, using separability of G, the following: 7.2.5
Lemma: If G is amenable, then there is a G.-invariant mean on UCB(G).
We now wish to pass from a mean on UCB(G) to one on L00(G). We shall do this by smoothing an L 00-function to obtain a uniformly continuous one. If cp, f are measurable functions on G, we recall that cp *f is defined to be the function (cp *g )(x) = J cp(g)f(g - 1 x)dg, as long as this exists for almost all x E G.. We recall some basic facts concerning convolution.
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Kazhdan's property (I)
7.2.6 Proposition: (a) IffE L00(G), cp E P(G), then CfJ *fE UCB(G) (b) Ifcpn , cp E L 1 (G) and (/Jn ---> cp in norm, then CfJn * f ---> cp *f in L 00 (G)for alljE L00(G). (c) For any G, there exists an approximate identity in L 1 (G), i..e. a sequence en E P(G) such that for all cp E L 1 (G), en * cp ...... cp and cp * en ...... cp in L 1 (d) For cp E e (G), fE L00(G), and g E G, CfJ * (n(g)f) = [A(g- 1 )(p(g- 1 )cp)] *f, where A is the modular function of G, p is the representation of G on functions given by (p(g)cp)(y) = cp(yg) (and n, is as above, the left regular representation). (e) (n(g)cp) * f = n(g)(cp * f).
If G is discrete, cp E P(G) and f E L "'(G), then cp * f =
cp(g) � 0, 'I cp(g) geG
=
'I cp(g)(n(g)f).
gEG
Since
1 , it follows that for an invariant mean m, we also have
m((p * f) = m(f). We shall need this property for arbitrary G, for the mean in Lemma 7 . 2 5 .
.
If m is a G-invariant mean on UCB(G), then for cp E P(G), /E UCB(G), we have m(cp * f) = m(f). 7.2.7
Lemma:
Proof: [Greenleaf 1 ] The result will follow horn our discussion of the barycenter construction following the statement of Proposition 4.. 1 A and a formal interpre
tation of the averaging effect of convolution as a barycenter construction. Namely, we first remark that it suffices to prove the result if supp(cp) is compact Fix such a cp and fix /E UCB(G). Define a map F : G ---> UCB(G) by F(g) n(g)f Since f is uniformly continuous, F is continuous.. Let 11 be the Haar measure on G and dv cpdJ1, so that v is a probability measure on G with compact support. Let K = supp(v)( = supp(cp)) . Then F(K) c UCB(G) is norm compact, and hence so is its convex hulL The measure F.(v) is thus a probability measure supported on a (separable) compact convex set in UCB(G).. Thus, we can let h b(F.(v)) be the barycenter, h E UCB(G). We recall that h is characterized by the condition that for all A E UCB(G) * , 2(h) = JA(y)d(F.v)(y).. We claim that in fact h = cp *f· To see this, for x E G, let Ax E UCB(G)* be evaluation at x Then =
=
=
.
..
h(x) = )ox(h)
=
J2x(y)d(F * v)(y)
J2x(F(g))dv(g) = Jf(g- 1 x)cp(g)dJ1(g) =
=
(cp * f)(x).
Ergodic theory and semisimple groups
1.38
To see that m(cp *f) = m(f), we can now simply observe that since
m E UCB(G) * ,
m(h) = Jm(y)d(F* v )(y) = Jm(F(g))dv(g) = Jm(f)dv(g)
by G-invariance of m
= m(f) We can now obtain an invariant mean on L 00 (G ).
then there exists a G-invariant mean m on L '"'(G) which satisfies the further condition that m(cp * f) = m(f)for all cp E P(G), fE L 00(G) 7.2.8
Lemma: If G is amenable,
Proof: Let m be an invariant mean on
UCB(G) (Lemma 7..2..5) . Fix any ljJ E P(G) and for f E L 00(G), define m( f) = m(ljf *f) . (This is defined by 7. 2. 6a. ) Since m is a mean, so is m. We claim that if cp E P( G), m(cp * f) = m(f).. Namely, if {en} is an approximate identity (7..2..6c), m(cp * /) = lim
m(cp Hn * /)
= lim m(ljf * cp Hn * f) =
lim m(en *f) by Lemma7.2 .7 and the fact that en *fE UCB(G) by 7. 2. 6a.
= lim m(ljf * en * f) for the same reason = m(ljf *f) = m(f). To see G-invariance of m, let g E G, fE L00(G). Fix cp E P(G). . Then m(n(g)f) = m(cp * n(g)f) by the preceding paragraph. By 7.2..6d, cp * n(g)f = rx *f where rx(x) = Ll(x) - 1 cp(xg - 1 ).. However, rx E P(G), and hence, m(n(g)f) = m(rx *f) = m(f) by the preceding paragraph, completing the proof With these preparations, we now proceed to the proof of Theorem 7.. 18 in general. Proof of Theorem 7.1.8: In light of Theorem 7.2.4, it suffices to show that if G is
amenable, then the regular representation almost has invariant vectors. Choose a G-invariant mean m as in Lemma 7..2. 8. By Proposition 7.2 ..3(6), there is a net CfJ i E P(G) such that m IR + is the Radon-Nikodym cocycle. (See the discussion pre ceding Proposition 4..2.20.) IfE c G/G;.0, and :YtE c L2(G/G;.0, :Yt0) is the subspace of functions supported on E, it follows that w(g)Yl'E g = Yl' E· Since we also have n(g):YtE 9 = Yl'E by Proposition 2.3.5 (after switching to a right action in 2.3.5), it follows that for all E c G/G;.0, n(g)w(g) - l Yl'E = ;YtE· However, if (M, Jl) is a (standard) measure space, and T is a unitary operator on L2(M, Jl, Yl'0) such that TYl'E = Yl'E for all E c M, then there is a Borel function M ·-> U( :Yto) m -> Um E U(Yl'o ), such that ( Tf)(m) = Um(f(m)). (See [Dixmier 2], for example.) Thus, for each g, we have [(n(g)w(g)- 1 )f](Jc) = a(Jc, g)f(Jc) for some a : G/G;.0 x G -> U(Yl'o ), and replacingf by w(g) f, we obtain (n(g)f)(Jc) = a(Jc, g)p(Jc, g) 11 2j(Jcg). Since n is a representation, it follows (as in section 4.2) that a is a cocycle
141
Kazhdan's property (T)
By Proposition 4..2J 5 there is a strict cocycle a' such that for all g, a(A, g) = a'{}o, g) a.e. It follows that a' corresponds to a representation 0' : G ;,0 --> U(Jfl' 0 ) by Proposition 4.2.1 3, and by the definition of induced representation, n = ind8.'·0(0'). Since inducing preserves direct sums and n is irreducible, (J must be as well To see the second assertion of the theorem, we observe that if g E IR", g acts trivially on lR", and hence on G/G i.o · Thus, for t E IR" we have (n(t )f)(A) = a'(A, t )f(A.) a e. for any /E L2(G/G;,0, Yl'o).. Identifying U(G/G;,0, Yl'o) with U(IR", fl, £ 0 ), we also have (n(t )f)(),) = A(t)f(A) by definition of n(JL, K,) Thus, for each t E IR", we have a'(A, t ) = A.( t) for almost all A (identirying A(t)EC as a scalar operator on Jfl'0 ). By Fubini, for 11-almost all A E lR" this equality is true of almost all t E IR", and since for fixed A, a'(A, t ) and A(t ) are continuous in t, for almost all A this equality holds for all t. (a' is continuous in t because a' is a strict cocycle, t acts trivially on lR", and hence t - > a'(A, t ) is a measurable homo morphism and hence continuous (Appendix B)). Fix Ao in the orbit supporting f1 such that a'(Ao, t) = ), 0 (t ) . By definition, 0' : G . U(Jfl' 0 ) is given by O'(g) = a'(Ao, g), and hence it follows that O' I IR" = (dim 0') Ao 7.3.2
Example: Let N be the Heisenberg group, i . e.
[
] [
]
Let A = { g E N i a = 0}, so that A � IR2. Then g E N acts on A by
1 0 z 0 1 .y 0 0 1
1 0 z- ay 1 y 0 0 1 If (a, [3) E IR2, so that A(a , pJ(y, z) = ei(ay + Pz) then identirying A with IR2 we have (g · A(a,p))(y, z) = A(a,p) (y, z - ay) = ei< IR2 where the action of SL(2, IR) on IR2 is given by usual matrix multiplication. The action on � 2 is just the adjoint action. Thus there are only two orbits of H acting on � 2, namely the origin and its complement If n is an irreducible unitary representation of H such that n I IR2 � n(Jl, f{',J with f.1 supported on the origin then n i 1R2 is trivial, and hence n factors to a representation of SL(2, IR). If f.1 is supported on the complement, the stabilizer of a point in the complement can be taken to be isomorphic to the Heisenberg group N (This can easily be seen by realizing H as a subgroup of SL(3, IR).. Namely, consider 3 x 3 matrices of the form 7.3.3
[ I �] g
o
oI 1
where g E SL(2, IR).. This group is clearly isomorphic to H in such a way that SL(2, IR) c H corresponds to matrices with b = c = 0, and IR2 c H to matrices with g = I.. The stabilizer of X E � 2, x(c, b) = eib, is readily seen to be N.) Thus in this case n = ind�(o') where a is an irreducible representation of N. Therefore we deduce that if n is any irreducible unitary representation of H, either n I IR2 is trivial, or n is induced from a representation of N. 7.3.4 Example: We can generalize 73.3 as follows. Let G = SL(2, IR) or PSL(2, IR) and suppose G acts on IRn by a rational representation such that the only fixed point under G is the origin. Let H = G1>< IRn . If n is an irreducible
unitary representation ofG then either n i iRn is trivial or n � indZ0(a) where H0 c H is an amenable subgroup. To see this, using Theorem 73.1 as in Example 7.3.3, and 4 L6(b), it suffices to see that the stabilizer in G of any point in � n except the origin, is amenable. But since there are no fixed points other than the origin, any stabilizer must be (the real points of) an IR-algebraic group of dimension at most 2 . Any such connected group is amenable, and thus the stabilizer is a finite extension of an amenable group, and hence is amenable (Corollary 4. 17). ..
We remark that the same arguments yield analogous results in 7..31-7.. 3.4 if IR is replaced by any local field of characteristic 0.. To apply results on semidirect products to showing that certain groups have Kazhdan's property, it will be useful to generalize somewhat the notion of a representation almost possessing an invariant vector. Namely, we think of almost possessing an invariant vector as almost containing the ! -dimensional
Kazhdan's property (T)
143
identity representation, and we will extend this definition by replacing the ! -dimensional identity by an arbitrary unitary representation. Let G be a locally compact group and n a unitary representation on a Hilbert space :ff, Let { v; I i = 1, . , n} be an orthonormal set in :ff , Define a function /n,{ v ,) : G ---+ M(n x n, C)(the n x n complex matrices) by (/n, {v;)(g))ij
= ( n(g)v ; I VJ )
We then call a function of the form /n , {v,J an (n x n) submatrix of n. If {vi} consists of a single element, then a 1 x 1 submatrix is of course j ust a matrix coefficient in the sense of chapter 2.. For (aij) E M(n x n, C) let 1 1 (ai1) 11 = max l aii l 7.3.5 Definition: (i) Let a, n be unitary representations of G . Let e > 0 and K c G compact We say that a is (e, K )-contained in n if for every n x n submatrix f of a, there is an n x n submatrix h of n such that 1 1 f(g) - h(g) 1 1 < e for all g E K (ii) We say that a is weakly contained in n and write a -< n, if a is (e, K ) contained in n for every e, K
Since for any unit vector ll n(g)v - v ll 2 = 2 - ((n(g)v l v) + (n(g - 1 )v l v ) ), it follows that the ! -dimensional identity representation is weakly contained in n if and only if n almost has invariant vectors in the sense of 71 1 . The notion of weak containment is due to [Fell 1 , 2] As an example, we can extend Theorem 71 . 8.
Proposition: (a) Suppose G is amenable. Let n be the regular representation. Then for any representation a of G, a -< oo n. (b) Suppose G is a group with regular representation n. If I -< oo ·· n, then G is amenable. 7.3.6
I, we have I -< n.. It follows directly from the definitions that this implies that I ® a -< n ® a, ie , a -< n ® a To prove the proposition, it then suffices to observe that n ® a � (dim a)n. However, the map U : L 2 ( G) ® :ff" --+ - L2 ( G, :ff") defined by [ U(f® v)](g) = f{g)(a(g)v) is easily seen to be a unitary equivalence of n ® a and the representation ii on L 2 (G, :ffa) given by (ii(h)cp)(g) = cp(gh), and it = (dim a)n. (b) For any cp E L2 (G), define Acp(g) = (n(g) cp l cp) , so Acp E L00(G). In the proof of Proof: (a) By 71 . 8, for the ! -dimensional identity representation
Theorem 7. 2.4, we began with the assumption that there was a sequence of unit vectors lji1 E L2 (G) such that 11 n(g)ljl1 - 1/11 11 2 ---+ 0 as j ---+ oo, uniformly on compact subsets of G, or equivalently A."'ig) ---+ 1 uniformly on compact sets of G. However, if we examine the proof, we see that in fact it suffices to find unit vectors ljl1 such that .:1."'/g) ---+ 1 only as elements of (U(G)) *, i e. , in the weak- * -topology in L 00(G). For if we have weak- * convergence, we can define cp1 as in 7.2.4 and deduce
E rgodic theory and semisimple groups
144
that (g ----> 1 1 n(g - 1 )
1 uniformly on compact sets
i
where Wni = Vni/ 1 1 Vni 1 . Since for each n, L 1 Vni 11 2 = 1 , it follows that for the function 1 E L00(G) we actually have l E co(A). However, 1 is clearly an extreme point of Lw(G)l , and hence is an extreme point of co(A). By our remarks above, this implies 1 E A, completing the proof We shall need the following two general results . .
7.3.7 Proposition: Suppose that H c G is a closed subgroup and that a,n are representations of H with a -< n . Then indZ(a) -< indZ(n).
y : G/H ----> G be a Bore! section with y(K) compact whenever K is compact (Appendix A). We have a cocycle a : G/H x G ----> H given by a(y, g) = y(y)gy(yg) - 1 . We recall that indZ(a) is defined by ((indZ(a)(g))f)(y) = p( y, g) 1 12 a(a(y, g))f(yg) where /E L2(G/H, £'11) and p is the Radon-Nikodym Proof: Let
derivative.. (See section 4..2.) IndZ(n) is of course defined similarly. For notational convenience, set indZ(n) = U ", and similarly define U" To show that any submatrix of U 11 can be approximated by submatrices of U ", it suffices by standard approximation arguments to show this is true for a submatrix of u a defined by n (n finite) orthonormal functions fi : G/H ----> Yl'a such that fi is compactly supported and f; takes on only finitely many values . Fix such f1 , . . . , In and let K c G be compact and t: > 0. .
145
Kazhdan's property ( I)
Choose Y c G/H compact such that [i = 0 on G)H - Y for all i. Then there is a compact set K' c H such that for y E YK - 1 and g E K, cx( y, g) E K '. Let M = max { 1 fi(GIH ) 1
i
spanned by
ro ,
UN GIH). i
1 } Let V c Yf be the finite-dimensional subspace a
Choose an orthonormal basis of V and let
qJ
be the
corresponding submatrix of CJ Let If; be a submatrix of rr such that 1 1 1/J (h) ({J(h) 1 < siM 2 n for all h E K '. Let W c Yf, be the space spanned by the ortho normal vectors defining If;. By mapping one orthonormal set to the other, we obtain a unitary mapping T: V--+ W An elementary calculation then shows that for any x, V E V,
2 I < CJ(h)x, v) - - I. Suppose n has no invariant vectors. Let £' 0 = { v E £'" l n(g)v = v for g E IR" } and £' 1 = £' � . Since IR" c G is a normal subgroup £' 0 and £' 1 are invariant under n( G), and we let n0 and n 1 be the corresponding subrepresentations. From the fact that n >- I, it follows that ni >- I for either i = 0 or 1 If n0 >- I, then (no I SL(n, IR)) >- I, and by Theorem 7 4. 2, there are no I SL(n, IR) invariant vectors in £' 0 , which implies n "?;_ I Suppose on the other hand, that n 1 >- I. Let n 1 = JEilnx where ny is irreducible (2.31).. Since n 1 I IR" has no invariant vectors, it follows that for almost all x, nx I IR" has no invariant vectors (2. 3..2). The action of SL(n, IR) on iR" has two orbits, namely the origin and its complement Applying Theorem 73. 1 to the representation nx, since nx I IR" has no invariant vectors (almost all x), it follows that for almost all x, the orbit that arises must be the complement of the origin. Therefore, for almost all X, nx = ind8a(ITx) where Go is the stabilizer in G of a non-0 point in !Rn However, Lebesgue measure on iR" is an invariant measure on G/G 0 , and since this measure is not finite, it follows that nx I SL(n, IR) does not have non-trivial invariant vectors . Thus n 1 I SL(n, IR) does not have non trivial invariant vectors. However, n 1 >- I, so (n1 1 SL(n, IR)) >- I, and by Theorem 7A..2, n 1 (SL(n, IR)) has non-trivial invariant vectors . This contradiction shows that n 1 >- I is impossible, and this completes the proof Proof: Let
[ Wang 1]: For n "?;_ 3, SL(n, Z ) t>< Z " has Kazhdan's property.
7.4.5
Corollary
Proof:
7 43 and 7AA.
8
8.1
Normal Subgroups of Lattices Margulis' finiteness theorem - statement and first steps of proof
The point of this chapter is to prove the following finiteness theorem of Margulis.
8.1 . 1 Theorem [Margulis 6]: Let G be a connected semisimple Lie group with finite center and no compact factors, and let r c G be an irreducible lattice Assume IR-rank(G) � 2 . Let N c r be a normal subgroup such that f'/N is ndt amenable. Then N c Z(G) (the center of G) and in particular N is .finite. .
Combined with Corollary 7. 1. 1 2 and Theorem 7. L1 3, this yields the following result
Theorem (Margulis-Kazhdan): Let G be a connected semisimple Lie group with finite center and no compact factors, r c G an irreducible lattice. Assume IR rank( G) � 2 . Let N c r be a normal subgroup. Then either 8.1 .2
(a) N c Z(G), and so N is finite; or (b) rjN is finite. Remark: We recall that while we proved Corollary 7.1. 1 2 in Chapter 7, we did not present a proof of 7.. 1 . 1 3.. Thus, after we prove 8. L 1, we shall have a complete proof of 8. . L2 when the IR-rank of every simple factor of G is at least 2 . The remaining case is complete once 7.. L 1 3 is proved, and we again refer the reader to [Margulis 7] for this proof. Earlier results in this direction were obtained by [Raghunathan 4] by different methods.. .
Margulis' proof of Theorem 8 . U is based on his discovery of a fundamental measure theoretic property of the action of r on G/P where P c G is a minimal parabolic subgroup . Namely, suppose P' => P is another parabolic subgroup. Then clearly there is a measure preserving r-map G/P -+ G/P'. Margulis dis covered that under suitable hypotheses, any r-space X for which a measure class preserving r-map G/P -+ X exists must be of this form .
1 50
Ergodic theory and semisimple groups
6]: Let G be a connected semisimple Lie group with no compact factors, trivial center, and IR-rank(G) ;;::; 2 . Let P c G be a minimal parabolic subgroup and f c G an irreducible lattice . Suppose (X, J.L) is a (standard) measurable r -space and that there is a measure class preserving f-map rp GIP --+ X (possibly defined only a . e.). Then there is a parabolic subgroup P' :=J P so that as r -spaces, GIP' and X are isomorphic in such a way that rp corresponds (a. e ) to the natural [-map GIP -+ GIP' ( We assume GIP, GIP' to have the G.-invariant measure class.) 8.1 .3
Theorem [M argulis
.
Let us see how this result implies the finiteness theorem
G by its center, we may assume that G is center free and wish to show that N is triviaL Since rIN is not amenable, there is a compact metric rIN-space X
Proof of Theorem 8.1.1 from Theorem 8.1 .3: With G as in 8. 1 . 1 , dividing
without an invariant probability measure . We may also consider X as a compact metric f-space (on which N acts trivially).. The f-action on GIP is amenable (Example 4 3. 8), and hence by Proposition 4.3 .9 there is a measurable r-map (defined a. e..) rp : GIP --+ M(X ) (where, as usual, M(X ) is the space of probability measures on X ). Let J1 be a probability measure on GIP quasi-invariant under G. Then rp * (J.L) is a measure on M(X ) quasi-invariant under r, and q; :(GIP, J.L) -+ (M(X ), rp * (J.L)) is a measure class preserving f-map . By Theorem 8.1 . 3, there is a parabolic subgroup P' :=J P such that (M(X ), rp * (J.L)) � GIP' as f-spaces. If P' = G, then r would have a fixed point in M(X ), which would contradict the choice of X. Therefore P' i= G. Recall that N acts trivially on X, hence on M(X ), and therefore each element of N fixes almost all points in GIP'. Since N acts continuously on GIP', this implies that N acts trivially o n GIP', so that N c Let G Then H
n gP'g - 1 , and H = n gP'g - 1 is a proper normal subgroup.
geG
geG
=
Il G i be the decomposition of G into a product of simple groups..
=
Il Gi, where J c I is a proper subset. Since N H, N is normalized
iel
c
i eJ
by Il Gi· · But since N is normalized by Il Gi and by r, it is normalized 1-J
I-J
b y the product of these groups which i s dense i n G b y irreducibility . Thus, N c G is normal, and since N is discrete and G has trivial center, N is triviaL This completes the proof. Before turning to the proof of Theorem 8. 1 3, it will be convenient to reformulate it slightly. We recall that if (X, J.L) is a measure space, by its measure algebra (or its Boolean a-algebra), B(X ), we mean the space of measurable subsets of X, two sets being identified if they differ by a null set. We can identify
Normal subgroups of lattices
!51
B(X ) as a subset o f the unit ball i n L "'(X ), namely B(X ) = { f E L aocx ) I f 2 = f(a e. )}. Then B(X ) is a weak-*-closed subset of L ao(X ). The induced topology on B(X ) is independent of the particular measure on X within its measure class. If
( Y, v) is a measure class preserving map, we obtain a map
B(X ) which is injective, and
B(X ) is an injedive continuous Boolean operation preserving map, then there is an a.e. defined measure class preserving map
( Y, v) such that
<J L, and that A is an automorphism of G which restricts to automorphisms of both H and L We shall assume that A - t is contracting on H and that A is the identity on L. Let E c G be a measurable subset We want to identify lim A"(E) in the spirit of 8.2.8. For each y E L, let Ey = {x E H l xy E E}. By 8.2 . 8, for each y, we have that for a.e . h E H, A"(hEy) --+ H (or 0) in measure on H if e E hEy (or e rj; hEy). We observe that e E hEy if and only if y E hE By Fubini, we have for almost all h E H, that for almost all y, A"(hEy) -+ H (or 0) in measure on H if y E hE (or y rf; hE). Once again using Fubini's theorem (by integrating over y) we deduce that for almost all h E H,
A"(hE) --+ H · (hE n L) in measure on G.. Suppose we have h E H for which this is true, and let y E L be arbitrary. Then yA"(hE) --+ yH (hE n L), and since A is trivial on L and y normalizes H, we deduce the following.
1 57
Normal subgroups of lattices
Corollary: Let G = H ><J L and A be as above. Then (possibly after replacing A by some positive power of A),for any E c G, we have that for almost all g E G, A n(gE) --> lj;(gE) in measure on G, where lj;(F) = H (F n L) for F c G. 8.2.9
8.3
Completion of the proof - equivariant measurable quotients of flag varieties
In this section we present Margulis' proof of Theorem 81 . 4.. As with the proof of the superrigidity theorems in Chapter 5, we shall use the fact that up to null sets, we can identify G/P with a unipotent subgroup of G (Lemma 5 1 .4).. While this was useful in chapter 5 for showing that a suitable measurable map on G/P was rational, here it will be useful as we will be able to apply the results of section 8 .2 on contracting automorphisms. We shall begin by reviewing some structural facts about semisimple groups (with the hypotheses in 81 .4), presenting SL(n, IR) as an example. In this section we shall be taking G/H to be {gH } , so that G acts on the left on G/H Let A c G be a maximal connected IR-split abelian subgroup (so for SL(n, IR) we can take A to be the positive diagonals..) Let P be a minimal parabolic subgroup containing A, and V the unipotent radical of A.. (Thus for SL(n, IR), we can take P to be the upper triangular matrices and V the subgroup of P consisting of all matrices in P with all diagonal entries equal to 1 .) Let P be the opposite parabolic, with unipotent radical V, (for SL(n, IR), the corresponding lower triangular matrices. ) Then (Lemma 5 . 1 .4), the restriction ofthe natural map G --> G/P to V defines a rational isomorphism of V with an open conull subset of .
G/P. Now let P0 ::::;, P be another parabolic. (For SL(n, IR), take P 0 to be the stabilizer of a flag as in the discussion preceding 8..24. ) Let V0 be the unipotent radical of P 0 , and P0 , V0 the corresponding opposite subgroups. One shows as in Lemma 5.1 4 that the natural map V0 --> G/Po is also a rational isomorphism onto an open conull subset of G/P0 .. In 8..14, we are interested in the map G/P --> G/P0 , and hence we wish to express this in terms of V, V0 In other words, we want to identify the map V--> V0 making the following diagram commute.
(* )
V
G/P
l(t Vo
G/Po
I I I I I I I
Let R 0 be the reductive component of
1
P
0 containing A, so P 0
=
R0 t>< V0,
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Ergodic theory and semisimple groups
P0 = Ro 1>< V0 (For SL(n, IR), this group is also described in the discussion preceding 8. 2.4.) Let Lo = R 0 n V( = Po n V) (For Po c SL(n, IR), as above, Lo = { g E R 0 I g;; = 1 and g;i = 0 if i < j }.) Then V0 , L0 c V, and in fact V = V0 ><J L0 Thus, the map V ---> V0 we needed above is simply the projection of V onto V0 given by this semidirect product decomposition. Let i : V ---> G/P be the natural map, which is a measure space isomorp hism For g E G, the action of g on G/P will then give us an a.e . defined measure space automorphism of V More precisely, if g E G, then for almost all x E V, gi(x) E i( V). Therefore, we define g o x E V via the equation g i(x) = i(g x), so that for each g this is defined a.e. For certain elements of G, we can be more explicit We observe that a
(i) If g E V, then g o x = g x for all x E V. (ii) If g E P n P (i. e. , g E R where P = R 1>< V), then g o x To see (ii), note that i(gxg - 1 ) = gxg - 1 P = =
=
gxg - 1 .
gxP gi(x).
By the equation defining g o x, we obtain (ii). We shall identify the G-action on B(G/P) with this induced action on B( V ). Now let B c B( V) = B(G/P) be a r-invariant closed Boolean s ubspace, and suppose E c V with E E B Let g E G. Of course, if this implied g · E E B, we would be done. Our first reduction of the proof is to show that it suffices to show that g ·· E E B where E is a certain subset related to E. M ore precisely, suppose E c V, and let P 0 -:::J P be as above, so that V = Vo ><J Lo Define t/lo(E) = Vo (E n Lo) (Cf 8. 2.9.).
To prove Theorem 8. 1 .4, it suffices to prove that for any V E E B c B( ), g E G and parabolic P 0 , G -:::J P 0 -:::J P, we have g · t/J0(vE)EBfor almost
8.3.1
Lemma:
-
all v E V
*
*
Proof: We continue to identify B( V ) � B(G/P). If Po -:::J P we shall also identify
B(G/Po) as a subspace of B(G/P) via the natural map GjP ---> G/P0 . In light of our discussion above, this identification is identified with the realization of B{ V0 ) as a subspace of B( V ) induced by the pr�jection V --+ V0 • Let B' be the largest G-invariant closed Boolean subspace of B, so we have B' c B c B(G/P). By our remarks in section 8J B' can be identified with B(G/P')
159
Normal subgroups of lattices
where P' ::::J P. We may have P' = G (so that B' is trivial), but we can assume P' #- P for otherwise we are clearly done. We claim that B' = B Suppose not, and choose E c G/P with E E B but E rj; B(G/P') . , Pm be the standard parabolics with P' ::::J Pi ::::J P, and Pi minimal Let P1 *
with respect to this property. As above, write Pi = R i 1>< Vi Li = R i n V The structure theory of semisimple group implies that P' is generated by {Pi} (This is a straightforward exercise for G = SL(n, IR) ) We have the map G ----> GjP, and we can thus identify B(G/P) as a subspace of B(G), and since E E B(G/P), we have, as an element of B(G), that E g = E for all g E P On the other hand, since E rf; B(G/P'), we cannot have E g = E for all g E P' Thus, for some i, there is an element g E P; such that E g #- E (modulo null sets). In other words, E E B(G/P) is not in the image of B(GjPi) under the embedding B(G/Pi) -> B(G/P) defined by the natural projection . In terms of the diagram ( * ) above, with P 0 = Pi, this means that considering E E B( V), we have E [; #- E. We now consider E E B{ V), and recall that V = Vi ><J [; Since E Li #- E, it follows from F ubini's theorem that for a set of w E vi of positive measure, { y E r; � wy E E} is neither null nor conull in L;, i e ' [; n w- 1 E is neither null nor conull in L;. For such a W E vi, we clearly have for any V E WLi that L; n v - 1 E is neither null nor conull in L; . Hence, we deduce that for a set of v E V of positive measure, 1/J i(vE) E B( V ) is not right I;-invariant, hence not ?'-invariant. In other words, under the identification in diagram ( * ), 1/J;(vE) is not in the image of B(G/P;) c B(G/P), and in particular is not in the image of B(G/P') (which is B '). However, IR-rank(G) � 2, and hence P; #- G . Therefore, by the hypotheses of the lemma, we can find v E V with 1/J;(vE) rj; B', but g 1/J ;( vE) E B for all g E G. Let B be the closed Boolean subspace generated by { gi/J;(vE) I g E G} and B' . Then B c B and B is G.-invariant However, 1/J;(vE) E B - B', so this contradicts the maximality assumption on B', proving Lemma 8.JJ . ,
The completion of the proof of Theorem 8 . 1 .4, i.e. , verification of the conditions in Lemma 8.3J, will follow from the following two lemmas . 8.3.2
Lemma:
Let P0 be a parabolic with G ::::J P0 ::::J P. Write Po = R o ><J Vo as *
above, and as in 8..2.4 , choose non-trivial s E Z(R0), the center of R 0 , (with s E A as in 8.2.4) such that Int(s) is contracting on V0 and Int(s) - ! is contracting on V0 Then for any measurable set E c V, we have (possibly replacing s by sN for some N � 1), for almost all v E V, as n ----> oo, Int(s)"(vE) ----> 1/!o(vE) in measure on V Proof: Apply Corollary 8..2.9.
The second lemma we will need is an ergodicity type of statement.
Ergodic theory and semisimple groups
160
8.3.3
Lemma: Let s be as in 8 ..3.. 2.
is dense in G
Thenfor almost all V E V, { yv- 1 s - n I y E r, n ;?; 1 }
This lemma is close to being a consequence of Moore's theorem (2 ..2. 6). Namely, the latter implies that the integer action defined by powers of s acts ergodically on G;r, and hence (by 2. 1 . 7), for almost all g E G, { ygs - n I y E r, n E Z } is dense in G. Thus 8. 3 3 differs from this last assertion in two respects. First, in 8.. 33 we consider only positive powers of s - 1 , and second we have an assertion about almost all v E V The latter is of course a lower dimensional space than G, and therefore such an assertion is not an immediate consequence of M oore's theorem. We shall show how to obtain 8 ..33 from Moore's theorem, but first show why 8 ..3.. 2 and 8.1.3 suffice to prove Theorem 8. 14. Proof of Theorem 8.1.4: We verify the condition of Lemma 8.J . L Let
E c V with
E E B.. Then for almost all v, the conclusions of 83.2 and 8..33 are valid . Fix g E G. Then by 8.3 . 3, we can write g = lim Yiv 1 s - n ; where ni ;?; 1. Furthermore, j
-
since the conclusion of 8. 1 3 is valid if we replace s by some positive power of
s, we can assume ni � oo as J � oo. Let gi = Yiv - 1 s - n;, so that Yi = gisn iv. Recalling that we are denoting the G action on V and B( V) by we have Yi o E = (gisn ;v) o E = gi o (sn ivEs- n ;) by (i), (ii) in the discussion preceding 8.3.. L As j � oo, gi � g and sn ivEs - n ; � t/J 0 (vE) i n measure o n V W e thus have Yi o E � g o t/J 0 (vE) in measure, and hence in the weak- * -topology on B( V).. However, since E E B and B is closed, we deduce that g o t/J 0 (vE) E B and by Lemma o,
831 , this completes the proof of Theorem 8. 14.
It therefore remains only to prove Lemma 8 . .3 ..3 We begin with a general observation to deal with the fact that we restrict ourselves to n ;?; 1 in 8.3.3 .. 8.3.4 Proposition: Suppose the group of integers Z acts ergodically and with a finite invariant measure on a space (S, /1) Then for any Y c S with Jl( Y) > 0, U ( Y n) is conull. n� 1
W = U ( Y· n). Then W 1 c W Since the action is measure preserving n� 1 and the measure is finite, W 1 = W. Thus W is essentially invariant under the Z-action, and by ergodicity W is conull Proof: Let
Normal subgroups of lattices
161
8.3.5 Corollary: With the notation of 8.34, if S is also a separable metric space and J1 is positive on open sets, then for almost all x E S, {x ( - n) I n � 1 } is dense in S
Proof: The proof of 2. L7 applies in this case as well.
W = {g E G I {ygs -" l y E r, n � 1 } is dense in G } . By Moore's theorem (2. 2. 6), {s-", n E Z } is ergodic on Gjr, and hence by Corollary 8. 3..5, W is conulL Recall that the multiplication map V x P ---> G is an injective map onto a conull subset of G. For each V E V, let Yv = { p E P i vp E W}, and let U = { v E VI Yv is conull in P} Then by Fubini, U is conull in V To prove the lemma, it suffices to show that e E Yv for all v E U Thus, we fix v E U Then we can choose Pk E Yv such that Pk ---> e .. By the choice of W, we then have that for each k � 1 , {yvpks- " l y E r, n � 1 } is dense in G, i.e , { yvs -"s"pks -" I y E r, n � 1 } is dense in G . Recall that we can write P = R r>< V, and that R is the centralizer of A in G. (In SL(n, IR), R is just the subgroup of all diagonal matrices.) Let Pk = ukrk, rk E R, uk E V Since s E A, s commutes with rk, and hence for each k � 1 we have {yvs -"s"uks -" I y E r, n � 1 } is dense in G. Since Pk ---> e, we also have uk ---> e. Suppose e fj; Yv, so that { yvs -" I n � 1 } misses an open set A c V Then we can find an open set B c A and an open symmetric neighborhood C of the origin in V such that BC c A . Thus B n (X - A)C = 0. However, { uk} is relatively compact since uk ---> e Thus, for n sufficiently large, Int(s)"( { uk}) c C since I nt(s) is contracting on V For any n, there are only finitely many elements of Int(s)"( {uk}) which do not lie in C, and it follows that for some k (in fact for all sufficiently large k), {Int(s")uk l n � 1 } c C But since {yvs -" l n � 1 } c X - A, and (X - A)C n B = 0, it follows that for some k, {yvs - "s"uks - " 1 n � 1 } does hOt inter sect B. This is impossible because this set is dense, contradicting the assumption that e fj; Yv This completes the proof of the lemma. Proof of Lemma 8.3.3: Let
9
9.1
Further Results on Ergodic Actions Cocycles and Kazhdan's property
If a group has the Kazhdan property, then there are significant restrictions on the type of cocycles certain actions of the group can have . We begin with the following
Theorem: Let G be a group with the Kazhdan property. Suppose S is an ergodic G-space with invariant probability measure. Let a .: S x G -+ H be a cocycle where H is an amenable group.. Then a is equivalent to a cocycle into a compact subgroup . (In particular, if H = IRn X z m , a is trivial.)
9.1 . 1
For G discrete, this was first shown in [Schmidt 2], [Zimmer 7J We preface the proof with the following general observation.
9.1.2 Lemma: Let G, H be locally compact, S an ergodic G--space and a · S x G -+ H a cocycle. Let n be the regular representation of H. Then a is equivalent to a cocycle into a compact subgroup of H if and only if there is a n o a-invariant function r.p . S -+ L 2(H h , the norm one vectors in L2(H ). (Definition 4.2J 7..)
a � f3 where f3(S x G) c K, K c H a compact group, then there is a n(K)-invariant vector in L 2(H ) t (simply because K is compact), and hence a n o {3-invariant function S -+ L 2(H ) t Therefore, there is such a n a-invariant function (4.2J 8c).. To see the converse, we first observe that for each r.p E L2(H ), n(h ) r.p -+ 0 in the weak- * -topology as h -+ oo in H. (To see this, suppose !/J E L2 (H ). Then we can find compact sets A, B such that 11 XACfJ - r.p 11 and 11 XB!/1 - !/I ll are smalL But as h -+ oo , we can assume Ah n B = 0, and hence 0. Let v denote Haar measure on G. If A c H is compact and g E G, let S(g, A) = { s E S i o:(s, g) E A, o:(s, g - 1 ) E A }
Since rx is measurable and H is a-compact, there is a compact set A c H such that (p,
x
v)( { s, g) E S
x K l o:(s, g) E A, rx(s, g - 1 ) E A }) ;?; (1 - e/3)(v(K)).
By Fubini's theorem, v(Ko) > 0, where Ko = { g E K i p,(S(g, A)) ;?; 1 - e/3 } .
Clearly K 0 is symmetric. Thus K 6 = K 0 K 0 1 contains a neighborhood W of the identity (Appendix B) By the cocycle identity o:(s, gh) = o:(s, g)o:(sg, h) and the fact that p, is G-invariant, for any y E K 6 we have p,(S ( y, A 2 ) ) ;?; 1 - 2£/1 Now choose g; E G, i = 1, . .. , n, such that u g; W => K. Then there is a compact set B c H such that p,(S(g;, B)) > 1 e/3 for all i . It follows as above from the cocycle identity that p,(S(g, BA 2 )) > 1 - t: for all g E K Since n almost has invariant vectors, we can choose a unit vector x E L 2 (H) such that 11 n(h)x x 1 1 < dor all h E BA 2 Define q; E L2 (S; L 2 (H )) by q;(s) = x for all s E S. Then for g E K, we have -
-
p,( { s E S I ll n(o:(s, g))q;(sg) - q;(s) 1 1 < t: }) > 1
-
£..
Therefore an elementary computation shows that 1 1 a(g)q; - q; 11 < A-(e) for all g E K where A-(e) -+ 0 as £ -+ 0.. This verifies our assertion that a almost has invariant vectors . Since G has the Kazhdan property, this implies that a has a non-trivial invariant vector. In other words, letting H act in L 2(H ) via n, there is an a-invariant function f: S -+ L 2 (H ) Finally, we remark that we can assume that for all s,f(s) E L 2 (H h by ergodicity. More precisely, since f =P O, f(s) =P 0 on a set of positive measure and since o:(s, g) f(sg) = f (s) a.e , ergodicity implies that 1 1 f(s) 11 is essentially constant The theorem now follows from Lemma 9. 1 . 2. We remark that this theorem is false if the assumption of finite invariant measure is eliminated .
E r godic theory and semisimple groups
1 64
Corollary: Let G = [1 G;, G; a connected simple non-compact Lie group withfinite center. Let S be an ergodic G-space withfinite invariant measure. Assume that G has the Kazhdan property, eg , !R-rank(G;) � 2 for all i. Suppose rx S x G -+ H is a cocycle with H amenable Then rx is equivalent to a cocycle into a compact subgroup of H (In particular, if H = !R" X zm, (X is trivial..) 9.1.3
Proof: This follows from 9. . 1 1, and 7. 1 4.
Kazhdan's property has significant implications for the kind of actions such a group can have on a compact manifold. (See [Zimmer 1 7].) The geometric background required for most of these results makes them beyond the scope of this book However, we shall be applying Theorem 9.1J in section 9.4 to study entropy of actions of semisimple Lie groups or their lattices on compact manifolds. Here, we only present a very elementary illustration of the geometric implications of the Kazhdan property, showing that groups with this property do not have volume preserving hyperbolic actions . We recall the latter notion Let M be a compact manifold and f: M -+ M a diffeomorphism.. There are a variety of conditions of f that describe "hyperbolic" behavior. Here, as an illustration we consider the following notion. See for example [Katok 1 ] for a discussion of notions of hyperbolicity. Call f weakly hyperbolic if there is a cif-invariant splitting of the tangent bundle of M into measurable subbundles, TM = E 1 + E2 such that for all v E E 1 , 1 drv 11 -+ 0 as n -+ oo and for all v E E2 , 11 dj - n v 1 -+ 0 as n -+ oo . Since M is compact, this is independent ofthe Riemannian metric If r is a group acting on M by diffeomorphisms, call the action weakly hyperbolic if the tangent bundle splits into dr-invariant subbundles as above, and with respect to this splitting, there is some y E r that is weakly hyperbolic. 9.1.4 Proposition: Suppose r is a group with the Kazhdan property Then r does not have any weakly hyperbolic volume preserving ergodic actions on a com pact manifold. Proof: Suppose we had such an action . We can measurably choose orthogonal bases for E;, and hence dj" acting on each E; will give us a cocycle rx; : S x r -+ GL(n;, !R), and for some Yo E r, and i = 1 or 2, 1 rx;(s, y�)v 11 ---. 0 for all v E !R"' . (Cf. Example 4..2.3.) Let us simply denote rx; by rx. Let P : S x r -. !R be p(s, y) = log l det rx(s, y) l . Then for all s, we have p(s, y�) -+ - oo as n -. oo. However, by Theorem 9. Ll, p is trivial, i.e., there is a function
(sg) = c.D(s) a(s, g).. In other words, c.D is an a-invariant function. As we have seen that Hk is smooth on E, the cocycle reduction lemma (5.. 2 . 1 1) applies and a � f3 where f3 takes values in the stabilizer of some element of E However, any such stabilizer clearly leaves a finite subset of H k/Lk invariant On the other hand, the stabilizer is the set of k-points of a k-group, and since the algebraic hull of a is H k, the stabilizer m ust actually be Hk Therefore, we deduce that Hk actually leaves a finite subset of Hk/Lk invariant, i e , Hk/Lk is finite. Thus, L :::J H 0, completing the proof
9.3
Actions of lattices and product actions
The rigidity theorem for ergodic actions (5..21) is one major respect in which ergodic actions of semisimple groups exhibit behavior markedly different from actions of amenable groups. The cohomology result (9J J) is another example of a marked difference between the semisimple and amenable cases In this section we wish to describe another such result
9.3.1 Theorem [Zimmer 1 3]: Let G be the real points of a connected simple algebraic fR-group, and r c G a lattice. Let S be an essentiallyfree ergodic r-space with finite invariant measure. Suppose, for i = 1 , 2, that C are discrete groups acting ergodically, with finite invariant measure, on (S;, Jl;), and that the r-action on S is orbit equivalent to the r 1 X r 2 -action on S 1 X S 2 Then either S 1 Or S 2 is .fi nite (modulo null sets).
Ergodic theory and semisimple groups
1 70
We have stated this for real groups but the same result is true if IR is replaced by any local field of characteristic 0 A similar type of result is also true for actions of G itself, as well as for certain actions of irreducible lattices in semi simple groups and for actions of certain discrete groups arising as fundamental groups of manifolds of negative curvature. We refer the reader to [Zimmer 1 3] for these and other generalizations. We also observe that this result stands in sharp contrast to the case when r is amenable . In fact, by 4.. 312 and 4.. 316, if r is amenable and acts properly ergodically and with finite invariant measure on S, then r is orbit equivalent to the r X f-action on s X S. As with many other results we have presented, our proof of 9.. 3. 1 will depend upon the behavior of measures on the variety G/P, where P is a minimal parabolic subgroup. In the proof of Theorem 9.31, we will need the following general result
r is a discrete group acting properly ergodically with invariant probability measure on (S, Jl.).. Then there is a discrete amenable group H which acts ergodically on S, preserves Jl, and such that for (almost) all s E S, sH c sf . 9.3.2
Proposition: Suppose
Proof: Since all our groups are countable, there is no problem in ignoring null
sets and we shall do so throughout the proof We first claim the following.. Suppose A, B c S where A, B are measurable disjoint sets and Jl.(A) = Jl(B) Then there exists a measure space isomorphism T: A -+ B such that for all x E A, x and Tx lie in the same f-orbit To prove this we define a sequence of subsets AN c A, BN c B inductively as follows . Let A 1 , B1 = 0. For N � 2, assume A 1 , . .. , AN - 1 , B 1 , .. , BN - 1 have been defined. Consider subsets D of A - A N - 1 of measure at least 1/N for which there is a measure space isomorphism T: D -+ E where E c B BN- 1 , with x and Tx in the same f-orbit. For a fixed N, there is clearly a maximal finite number of such triples (Di> Ti Ei) for which all Di> Ei are mutually disjoint Choose such a maximal collection of triples and let AN - 1 = uDi> BN - 1 = uEr Define AN = AN - 1 u AN - 1 , BN = B N - 1 u BN - 1 · Let A oo = u AN, Boo = uBN . By our construction of AN, BN, we clearly have the existence of a measure space isomorphism T: A ro -+ Boo with x, Tx in the same f-orbit If Aro = A, then we are done If not, let D c A - Aro with Jl.(D) > 0. Since Tis measure preserving Bro #- B, and we can choose E c B - B ro of positive measure . Since r acts ergodically D · r => E, and hence there is a subset Do c D of positive measure and y E r such that Do y c E However, if N is such that 1/N < J1.(D0), this contradicts the maximality of AN This shows A = A ro , verifying our assertion . Before proceeding, we introduce some terminology. If A c S is measurable, P = { B 1 , . . , B, } is a partition of A into disjoint measurable sets B; of equal -
>
Further results on ergodic actions
171
measure, and F i s a finite group o f measure space isomorphisms of A onto itself, let us say that F is ?-admissible if: for all TE F and all i, TB; = Bi for some j; F acts simply transitively on the elements of P; and for all x E A and TE F, x and Tx are the same r -orbit. The assertion of the preceding paragraph implies that for any partition P of A c S into r sets of equal measure there is a Z/rZ-action on A which is ?-admissible. If D c A is a measurable set and P is a partition of A, we let d(D, P) = inf {,u(D�B) / B is a union (possibly empty) of subsets belonging to P}. We remark that if D, E c A, then d(D,
P) � d(E, P) + ,u(D� E).
Now let {A;} be a sequence of measurable subsets of S which are dense in the measure algebra of S, and such that for each i, A; = A n for infinitely many n > i. We claim that we can inductively construct for each i the following: ( 1 ) a partition P; of S into 2"' mutually disjoint measurable subsets of equal measure, for some n; � i ; (2) a ?;-admissible action on S of a finite abelian group F;; such that (3) P; + 1 ::::J P; (i . e. P; + 1 is a "finer" partition); (4) F; + 1 ::::J F;; (.5) d(A;, P;) < 1/2 ; For i = 1 , any partition into two sets of measure 1/2 suffices . We now construct P;+ 1 from P; Let P; = { D 1 , , D 2 n.} We can find a number m > n; and for eachj a partition Qi= {Cid of Di such that: (6) ,u( C ik) = lj2m for all j, k; and (7) for each j, d(A ; + 1 n D h Qi) < 2 - ( i + Z + n,) We can then choose a partition R of D1 such that: (8) for some n; + 1 > m every set in R has measure 2 - (n, + ! l ; and (9) if Ti E F; is the unique element with Ti(D t ) = Dh then d(Tj 1 (Cik), R ) < 2 - (i + Z + n , + m) for any j, k. Define P; + 1 to be the partition of S into sets of the form T(E), where E E R and TE F;. Since F; is measure preserving, P; + 1 is a partition of S into 2"' + ' sets of equal measure. It follows from (9) that for all j, k, d( Cik, P; + t ) < 2 - U + 2 + n , + ml, and hence for any set E c D i which is a disjoint union of elements of Qi that
From this inequality and (7) we deduce that for all j,
Thus d(A ; + 1 , P; + t ) < 2"'2 - (i + 1 + n,) =
2 - (i + l J
This establishes condition (.5). To
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complete the induction, we construct a P; + 1 -admissible finite abelian group F; + 1 acting on S such that (4) is also satisfied. By our earlier remarks there is an R-admissible action of Z/rZ on D 1 where r = 2"' + ' - "' Let X E Z/rZ, j > 1, and Ti E F; be as in (9). Let X act on Di by Ti)Jj 1 This extends the ZjrZ action to all of S and clearly Z/rZ commutes with F;. Thus, letting F i + 1 be the group generated by Z/rZ and F;, the induction argument is complete Let H = u F; . Then H is abelian and hence amenable . Furthermore sH c sr and so to prove the proposition it suffices to see that H acts ergodically on S. Suppose A c S is H-invariant Since F; c H transitively permutes the elements of P;, it follows for B, C E P;, we have .u(A n B) = .u(A n C).. Therefore ,u(A n B) = 2 - n, .u(A) = ,u(B).u(A).. It follows that .u(A n B) = .u(B),u(A) for any set B which is a union of elements of uP; . From (5) and the fact that A; = An for infinitely many n > i, we deduce that for each i, .u(A n A ;) = .u(A;),u(A).. Since { A;} is dense in the measure algebra of S, we obtain .u(A n B) = .u(B),u(A) for all measurable B. Hence, letting B = A, we conclude that A is either null or conull, establishing ergodicity. We now turn to the proof of 9 J J .
Proof o f Theorem 9.3.1 : Throughout this proof� "algebraic group" will mean the set of real points of an algebraic IR-group. If S; is not finite, then 1; acts properly ergodically on S;. Choose an amenable subgroup A; acting on S; as in 9. 3.2, i e ' with xA; c xr; for X E S; Let 8: s --+ St X s2 be the orbit equivalence of the r-action with the r 1 X r 2 action . Let ex : (St X S2) X r 1 X r 2 --+ r c G be the cocycle corresponding to the orbit equivalence 8 - l (Example 4 .2..8), i.e. , for X E s X Sz, h E r 1 X r 2 , 8 - 1 (x)ex(x, h) = 8 - 1 (xh). By Theorem 9.2.. 3 (and Example 1 4. 2. 1 8), there is an amenable algebraic subgroup H c G and an ex I S 1 x S 2 x A t x Arinvariant function cp : S 1 x S2 --+ GjH. As the following lemma shows, to provide the desired contradiction, it actually suffices to show that there is such an ex-invariant function.. .
Lemma: To prove the theorem, it suffices to show that there is a proper algebraic subgroup L c G and an ex-invariant function cp : S 1 x S2 --+ GjL.
9.3.3
Let "' ; s --+ GjL be "' = cp 8.. For s E S, ')! E r, h E r 1 X r 2 such that 8(s) h = 8(sy), we have by the definition of ex that ex(8(s), h) = y, and hence that
Proof:
0
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Further results on ergodic actions
tjf(sy) = cp(B(sy)) = cp(B(s) h) = cp(B(s)) ex(B(s), h) [by ex-invariance of cp]. = t/J (s) Y In other words, t/1 : S --+ G/L is a 1 -map . If J1 is the 1 -invariant measure on S, then t/J*(Jl) will be a [-invariant probability measure on G/L We can then define a G-invariant probability measure on G/L as follows. Let m be the G-invariant measure on G/1 For A c G/L measurable, (t/J *Jl)(Ag - t ) depends only on the image of g in G/f, and thus we can define v(A) = j (t/J*(Jl))(Ag- t )dm. This is then clearly a G-in variant probability measure on G/L However, since L is a proper algebraic subgroup, this is impossible by the Borel density theorem (3.2..5) . This proves the lemma. Therefore, we need to pass from the existence of an a i St x S 2 x At x A 2 invariant function to the existence of a suitable ex-invariant function. We shall do this by developing a uniqueness property for a ! S t x S 2 x A t x A z-invariant maps. Suppose that for each s E S t we have an algebraic subgroup H, c G such that {(s, g)j gE H,} is a measurable subset of St x G. Let p, : G --+ G/H, be the natural projection. Let cp : S 1 x S 2 --+ G be a Borel map. Let us say that ( cp, { H,}) is ex-admissible if for almost all s E S 1 , p, o cp is an ex ! { s } x S 2 x A z-invariant function . For s E S 1 , h E 1 1 , x E S 2 , and a E A 2 , we have from the cocycle relation that
ex(s, x, h, e)ex(sh, x, e, a) = ex(s,
x,
e, a)a(s, xa, h, e). For a fixed h E 1 1 , and ex-admissible (cp, {H,}), define h *(cp, {H,}) to be the pair (t/1, fi,) where t/J(s, x) = cp(s · h, x) · ex(sh, x, h - t , e) and fi, = Hsh· The above consequence of the cocycle relation is readily seen to imply that h * (cp, H,) is also ex-admissible for any h E 1 1 Furthermore, it is clear that if (cp i , { H�}), i = 1, 2, are ex-admissible, and S 1 = B u C is a di�joint decomposition into measurable sets, then defining cp to be ({J 1 on B x S 2 and q>z on C x Sz, and
defining H, similarly from H �, we still obtain an ex-admissible pair. It follows from these remarks and ergodicity of the r 1 -action on S 1 that there exists an a-admissible (cp, {H,}) such that dim H, is essentially constant over s E S 1 and that for any admissible (t/1, { J,}), we have dim H, � dim J, a.e. The following observation is basic to the proof of Theorem 9.3. 1 .
Lemma: Suppose (cp, {H,}), (t/1, {J,}) are ex-admissible and both satisfy the above minimality of dimension property.. Then (i) H? and J ? are conjugate for almost all s.. (ii) The conjugacy class of H ? is essentially constant in s. In particular, we can 9.3.4
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choose H, satisfying the minimality of dimension property with H ? = H 0, independent of s. (iii) With such a choice, let p : G -+ GjN(H 0), where N(H 0) is the normalizer of H 0 (and is therefore algebraic). Suppose we also have J ? = H 0 a e , so that H ,, ], c N(H 0). Then p o cp = p o lj; Proof: Consider the map (cp, lj;): s 1 X s 2 -+ G For each s E Sj, let p, : G -+ GjH,, q, : G -+ GjJ, be the projections. Then (p, o q;, q, o rf;) yields an a l {s} X s2 X A 2invariant map {s} x S 2 -+ GjH , x Gjl, . Since A 2 is ergodic on S2, and G-orbits are locally closed in G/H, x G/1, (3. 1.3), it follows from the cocycle reduc tion lemma (5 ..21 1 ) and Example 4.2. 1 8(b), as in the proof of 9.2J, that (p, o cp, q, o lj;)({s} x S 2 ) is (a.e.) contained in a single G-orbit in GjH, x GjJ,. The stabilizer of such an orbit can be chosen to be of the form L, = H, n bJ ,b- 1 for some b E G. It is a technical exercise to see that s -+ L, can be chosen measur ably, and hence ((cp, lj;), {L,} ) is also a-admissible. By the dimension property, dim L, = dim H, = dim (J,) a. e. , and hence H ? and J ? are conjugate. To see that the conjugacy class of H ? is independent of s E S 1 , observe first that if h E r 1 , h * (cp, { H,}) will also satisfy the minimality of dimension condition, and so by the above H? and H ?h are conjugate. We now employ an argument of [Auslander-Moore 1]. Let L(H,) denote the Lie algebra of H,, so that L(H,) c L( G) is a linear subspace, and the dimension of this subspace is essentially constant for s E S 1 , say of dimension d. Since H ? and H?h are conjugate, it follows that L(H,) and L(H,h) are in the same Ad( G)-orbit for the action of Ad( G) on the Grassmann variety V of d-planes in L(G). However, the action of Ad( G) on V is smooth (3.. 13), and thus V/Ad(G) is countably separated.. Since L(H,) and L(H,h) are equal when projected to V/(Ad(G)), the map s -+ (pr�jection of L(H ,) in V/Ad( G)) is an essentially r � -invariant function on an ergodic r 1 -space taking values in a countably separated Bore! space. It follows that this map is essentially constant, or equivalently, on a conull set in S 1 , all L(H,) are in the same Ad(G)-orbit This implies that on a conull set in S 1 , all H ? are conjugate in G . The usual correspondence of cosets of a subgroup and that of a conjugate subgroup then enable us to choose an admissible (cp, {H,} ) satis(ying the minimality condition with H ? = H 0 a. e. Finally, suppose we also have J ? = H 0 a.e. As above, the image of the map (cp, lj;) will lie in a single G-orbit, with the stabilizer of a point in the orbit of the form L, = H, n bJ,b - 1 , for some b E G Since dim H, = dim J, = dim L,, bH 0 b - 1 = H 0, i.e., b E N(H 0 ).. Thus the image ofthe relevant orbit of GjH, x GjJ, in G/N(H 0 ) x GjN(H 0 ) contains a point on the diagonal, and hence is contained in the diagonal. Therefore, the image of (p o cp, p o lj;) is essentially contained in the diagonal, and this proves the final assertion of the lemma.
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Further results on ergodic actions
9.3.5
Lemma:
Let H, as in the conclusion of Lemma 9. 3.4. Then H0 i= { e } .
Proof: If it were, each H, would be finite Fix s E S 1 so that all required condi tions which are known to hold a.e . hold on almost all { s } x S2 In particular, qy , : S2 ---+ G/H, given by qy,(x) (p, o qy)(s, x) is an a j { s} x S2 x A z-invariant function. We can choose a compact set B c GjH, such that
00 in r, such that e - 1(x), e - 1(x)yn E 8 - !(qy; 1 (B)). Let lj; = qy o 8 Then the same calculation as in the beginning of the proof of Lemma 9. 3. 3 shows that 1 1 lj; (8 - 1(x) y n) = lj; (8 - (x) )Yn · However, e - 1(X) }'n E 8 - 1(qy; (B)), so lj;( 8 - 1(x) y n) E B for all n . O n the other hand, since H , c G is finite, if Y n E r, }'n ---+ 00 , then for any y E GjH,, YYn ---+ oo, i e , leaves every compact subset of G/H, In particular, this is true for y = lj; (8 - 1 (x) ), which is a contradiction . This proves Lemma 9.3 .5. =
We now complete the proof of Theorem 9 . 1 1 As in Lemma 9.3.4 choose (qy, {H,}) a-admissible, satisfying the minimality of dimension condition, and H� = H 0 As above, let h E f1 As observed preceding Lemma 9. 3.4, h*(qy, {H,}) will also be a-admissible It follows from the final conclusion of Lemma 9.3.4 and the definition of h*(qy, {H,}) that p(qy(sh, x)) a(sh, x, h, e)) = p(qy(s, x) ).. In other words, p o qy : S1 x S2 ---+ G/N(H0) is an a j S1 x { x} x f 1 -invariant map for a. e. x E S2 By our construction at the beginning of the proof, H 0 =1- G, and by Lemma 9. 1 5, H 0 i= { e } , and so N(H0) =1- G.. We can now repeat our entire argument working with a I S 1 x { x} x r 1 -invariant functions . Lemmas 9.3 . 4 and 9.3..5 hold in this situation as well, and repeating the arguments immediately above using }' E f 2 rather than h E f 1 , we deduce the existence of an a-invariant map qy : S 1 x S2 ---+ G/L for some non-trivial proper algebraic subgroup L c G. By Lemma 9.3.3, this completes the proof. 9.4
Rigidity and entropy
In Section 52 we applied the superrigidity theorem for cocycles (5. 2..5) to the cocycles arising from an orbit equivalence to deduce the rigidity theorem for ergodic actions . Theorem 5.2 ..5 can also be applied to the derivative cocycle for an action on a manifold (Example 4.2.3) or to other associated geometrically defined cocycles . This is a basic step in proving results about actions of semi simple groups or their lattices on compact manifolds . The full proofs, however, require some significant geometric background and so we shall not present these results here.. As an introduction to this subject, we shall present an application of superrigidity to the computation of entropy, a basic measure theoretic invariant of the diffeomorphisms defined by single elements of the
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Ergodic theory and semisimple groups
acting group. This particular consequence of superrigidity was first observed in [F urstenberg 5]. For further developments of the applications of superrigidity to actions on manifolds, we refer the reader to [Zimmer 1 5, 1 8, 1 9].. We shall first review the concept of entropy for an integer action . We will not provide proofs, but refer the reader to [Billingsley 1] and [Ornstein 2].. We assume that (S, Jl) is a Z-space, where J1 is a Z-invariant probability measure. Let T: S --+ S be given by Ts = s 1 . To this transformation, we will associate a non-negative real number h(T), called the (Kolmogorov-Sinai) entropy of T We begin by seeking a quantitative measure of randomness. Namely, suppose p = (p 1 , . . , pk ) is a probability measure on { 1 , . , k}, ie , 0 ;;:; p; ;;:; 1, 'L p ; = 1 . W e seek a number H(p), which we want t o measure the randomness of p, satisfying the following axioms: (i) H is continuous in p, H(p) � 0, H is symmetric, and H(p, 0) = H(p) (ii) H(l, 0, , 0) = 0. (No randomness.) (iii) For each k, H(p1, . , Pk ) achieves a maximum at p; = 1 /k for all i. (iv) If (p 1 , , Pk ), (q 1 , . . , qk ) are probability vectors, then We then have:
9.4.1
Theorem
[Khinchin 1]: Up to a scalar multiple, the only function satisfying
(i}-{iv) is H(p 1 , . . , Pk) =
- 'L
p ; log p ;.
H( p) is called the entropy of p We now let (S, Jl) be a standard measure space, J.l(S) = 1 . By a partition d of S we mean a collection d = { A 1 , . . . , Ak }, where A; c S are mutually disjoint measurable sets and u A; = S Define H(d) = H(JL(Al ), . . , Jl(Ak ) ). If d, f!J are two partitions, let d v !!4 be the partition d v !!4 = {A; n Bi }. It is then easy to check from property (iv) above that H(d) ;;:; H(d v !!4) ;;:; H(d) + H(f!J). If d = !!4, then clearly H(d v !!4) = H(d). Hence if !!4 is close to d, H(d v !!4) will be close to H(d).. On the other hand, suppose d and !!4 are independent partitions, i e , Jl(A; n Bi) = Jl(A;)Jl(B il (for example, partitions from partitions in factors of a product space.) Then we have H(d v !!4) = H(d) + H(f!B). Recall the map T: S --+ S The entropy of T will measure the rate of growth of H(d V T(d) V . . V r(d))..
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Further results on ergodic actions
9.4.2 Definition: (i) Set h( T, d) = lim H(d V T(d) V . . . V T" - 1 (d))/n. (ii) Set h( T) = sup {h( T, d) I A is a partition of S}. h(T) is called the entropy of T (or of the Z-action), and is clearly an invariant of the Z-action up to conjugacy . One can compute h( T) because of the following:
(Sinai): If d is a generating partition for the Z-action, i.e., the smallest closed Boolean subalgebra of the measure algebra B(S) that contains d and is Z-invariant is B(S) itself, then h(T) = h(T, d)..
9.4.3
Theorem
9.4.4
Example:
(a) Let (X, p) be a finite probability space. Let Q
=
00
I (X, p),
- oo
and define T: Q --> Q by (Tw)n = Wn + 1 This is called the Bernoulli shift o n the state space (X, p).. Then h(T) = H(p).. (In particular, h(T) can take any positive value. ) (b) If T is a rotation of the circle, then h(T) = 0 (c) Let A c SL(2, lP) be the positive diagonal matrices, and r c SL(2, lP) a cocompact lattice. For a E A, let h(a) be the entropy of a acting on SL(2, IP)jr Then up to scale, h(a) = log 1 1 a 1 1 . Thus, h(T) can take any non-negative value for T a diffeomorphism of a compact manifold. We now state two fundamental results that describe the relationship of entropy to Bernoulli shifts .
[Ornstein 1]: Any two Bernoulli shifts with the same entropy define conjugate Z-actions. 9.4.5
Theorem
[Sinai 1]: Suppose S is a Z-space with entropy h. Then there is a Bernoulli shift with entropy h, say on Q, and a measure preserving Z-map S --> n
9.4.6 Theorem
For a C 2 -diffeomorphism, the entropy may be computed from properties of the derivative cocycle on the tangent bundle . To describe this result, we first describe some general results concerning the asymptotic behavior of cocycles of Z-actions. If A is any n x n matrix, then lim *log + 11 A" 11 exists and is equal to max {log + I A- l l
A is an eigenvalue of A} If (S, p) is a Z-space, and a : S x Z --> GL(n, C) then a(s, n) = a(s, 1 )a(s 1 , 1 ), . . , a(s · n - 1 , 1 ), and so is naturally a product of n (unequal in general) matrices . For cocycles which are sufficiently regular, one can obtain
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Ergodic theory and semisimple groups
a similar result on the behavior oflog + ll a(s, n) 11 Namely, let us call a cocycle IX "tempered" if I l a( , n) 11 E L "'(S, fl) for each n.. Define (e(a) )(s) = lim * log + ll a(s, n) 11 n � eo
if it exists
Theorem [Furstenberg-Kesten 1]: Let (S, Jl) be a Z-space where f1 is an invariant probability measure. Suppose IX : S x Z ---> GL(n, C) is a tempered cocycle Then (i) e(a)(s) existsfor almost all s E A, and e(a) is essentially Z-invariant. Hence, if Z acts ergodically, e(a) is constant. (ii) If IX and [3 are tempered and IX � [3, then e(a)(s) = e([J)(s) (a. e.).
9.4.7
For a proof of 94..7, see [Kingman 1] and [Derriennic 1]. A basic result on the entropy of difleomorphisms is the following theorem of Pesin If the integers Z act on a compact manifold M by diffeomorphisms, compactness implies that 11 df� 11 is bounded over x E M for each fixed n (for any choice qf Riemannian metric) and hence that the derivative cocycle (Example 4..2..3) is tempered (since we can measurably trivialize the tangent bundle with orthonormal sections). Theorem [Pesin 1]: Suppose the integers act on a compact manifold by 2 C - diffeomorphisms so as to be volume preserving. Set Ts = s· L Let IX be the derivative cocycle, and let AP(a) be the p-th exterior power of IX Then the entropy
9.4.8
is given by h(T) = J[max e(N(a))(s)]ds In particular, in the ergodic case p
h(T) = max e(N(a)). p
Pesin's result depends in part upon a sharpening of 94. 7 due to [Oseledec 1] called the "multiplicative ergodic theorem" Before describing this, we make some remarks about a single matrix A E GL(n, IC) Let A;, i = 1 , . . . , k be the distinct real numbers of the form log + I ) I where A is an eigenvalue of A, and assume A1 < A2 .. < Ak Then there are subspaces 0 c V 1' c V2 c . . c Vk = IC" 0
*
such that for v E V; but v � V; - 1 , we have lim Mog + II A"v ll
=
Ai ··
Oseledec's
theorem generalizes this spectral type decomposition to tempered cocycles. Theorem [Oseledec 1]: Let (S, /1) and IX be as in 94.. 7 and assume the Z action is ergodic. Then there exist A 1 , . . , )ok E lP, A1 < ) 2 < . . . < Ak, positive integers 1 � n 1 < nz < < nk = n, and a-invariant measurable maps Vi : S ---> Grn,(C"), the
9.4.9
0
1 79
Further results on ergodic actions
latter being the Grassmann variety of n;-planes in en, with V;(s) c V; + 1 (s) such that for almost all s E S, we have for all v E V;(s), v i$ V; - 1 (s) lim � log+ ll o:(s, n)v ll = A; n - OCJ n Remark:
Clearly e(o:) = },k We also have max e(AP(a)) = �(n; - n; - 1 )).; The ),k p
are called characteristic (or Lyapunov) exponents of a. For a more recent proof of 9.4 . 9, see [Raghunathan 2] Example: If o:(s, n) = An where A E GL(n, C), then Theorem 94. 9 reduces to the observation we made following 9.4..8. If we know that a tempered cocycle a takes values in an algebraic subgroup of GL(n, C), we can say more about e(a). For example, if a takes values in a unipotent subgroup, then e(a) = 0 (as do all the characteristic exponents) . More generally, if G c GL(n, q is a connected algebraic subgroup, we can write G = R ex. U where U is the unipotent radical and R is reductive. For a cocycle a taking values in G, let o:R be the reductive component, i.e , ClR = p a where p : G � R is the projection. Then we have e(a) = e(o:R). We shall need the follow ing generalization of this remark 9.4.10
o
Proposition: Suppose a is a tempered cocycle taking values in GL(n, IR). Let H c G L(n, q be an IR-group. Suppose H is connected, and write H rK = R ex. U where R is reductive and U is unipotent. Let f3 be a cocycle equivalent to a with f3(S x Z) c HYK and, as above, let /3R be the composition of f3 with projection onto R. Finally, assume f3R is also tempered. Then e(f3R) = e(a). In concluding these preliminaries, we mention that the entropy and the characteristic exponents of a diffeomorphism have important consequences for the study of classical dynamical properties of the diffeomorphism, for instance periodic points . See [Bowen 1] and [Katok 1] for example . Now let G be a connected simple IR-group . Let G be the algebraic universal covering group, and n : GYK � SL(n, IR) an irreducible rational representation . If g 1 , g 2 E GrK and g 1 = g 2 in GrK, then g 1 = g 2 z where z E Z(GrK) (the center of GrK). Since Z(GrK) is finite, it follows that 9.4. 1 1
max { I A I I A an eigenvalue of n(g 1 ) } = max { I A I ! ). an eigenvalue of n(g 2)} .
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Thus, for g E G11, max { I X I I ), an eigenvalue of g} is well defined where g E G11 is any element projecting to g. We now present an application of the superrigidity theorem for cocycles (5 ..2..5). Combining 5.2 5 and 9. 4 . 8, we obtain the following 9.4. 1 2 Theorem: Let G be a connected almost simple fR-group with fR-rank(G) � 2. Suppose Gg acts by C 2 volume preserving diffeomorphisms on a compact manifold M, and suppose the action is ergodic. For g E G�, let h(g) denote the entropy of the diffeomorphism defined by the action of g on M. Then either h(g) = O for all g, or for each g E G�, there is an fR-rational representation n : G ---> SL(r, C) for some r, such that
h(g) = max {log(l A I) I A is an eigenvalue of n(g) } . Further, if n = dim M, n is a n exterior power of a representation a of G , where dim (J � n. Corollary: For a fixed n, and fixed g E G�, the set of possible values ofh(g) over all volume preserving, ergodic, C 2 -actions of G� on compact manifolds of dimension at most n is .finite. 9.4.1 3
Proof of 9.4.1 2:
Let a : M
x
G� ---> GL(n, fR) c GL(n, C) be the derivative cocycle.
For g E G� let a9 = a I M x { gn } . Thus h(g) = J max e(N(a9) ) by Pesin's theorem. p
Now let H11 c GL(n, fR) be the algebraic hull of a. As in 9.2.6, let S = M x aH11/(H 0)11, and define f3 = a as in 9.2. 6 as well. Since H11/�H 0)11 is finite, S has a finite invariant measure. Clearly for any p, e( N(a9) ) = e(AP(f39)). By 9 2. 6, the algebraic hull of f3 is (H 0)11. Let H0 = Lr>< U where L is reductive, U is unipotent, and L and U are fR-groups . Let y = f3L so that y : S x G� ---> L11 is a cocycle with algebraic hull L11 Let q : L11 ---> L11j[L, L]11. Then the algebraic hull of q y is LR/[L, L]11, and hence by Corollary 9. 1. 3 L11/[L, L]11 is compact Since L = [L, L] ·· Z(L) where Z(L) is the center of L, and [L, L] n Z(L) is finite, it follows that Z(L)11 is also compact We can write the fR-group L/Z(L) as a product of semisimple fR-groups L/Z(L) = L 1 x Lz such that (L2)11 is compact, and (Lt)R is center free and with no compact factors.. Let q l : L11 ---> (Lt)11 be the pr�jection. Then q1 y : S x G�-> (L 1 )11 is a cocycle with algebraic hull (Lt)R, and by the superrigidity theorem for cocycles (5.. 2 .5), there is a rational homomorphism n : G ---> L 1 defined over IR o
o
181
Further results on ergodic actions
such that q 1 y � IX, 1 a � where IX, i s the cocycle a,(s, g) = n(g). Thus, w e can write, for g E G2 and a. e. s E S, cp(s)q l (r(s, g))cp(sg) - 1 = n(g) where cp : S -> (L l )R is Bore! We can choose a Bore! section[ : (LdR -> LR of q 1 Then q 1 (f(cp(s))y(s, g)f(cp(sgW 1 ) = n(g). In other words, replacing }' by an equivalent cocycle 6 : S x G� --> LR, we have q 1 (6(s, g)) = n(g) We can consider n as a homomorphism n : G -> L 1 , where G is the (algebraic) universal covering group of G Then we can lift this to a homomorphism ii :G -> [L, L] c GL(n, C) defined over IR. Thus, for each g E G�, we have q 1 (6(s, g)) = q 1 (ii(g)) for almost all s, where {j E GR projects to g under the covering map. However, since (L2)R and Z(L2)11 are compact, (ker ql)R is compact Thus, ii(g) = b(s, g) b where b is some element of the compact normal subgroup (ker q l )R of LR It follows that for any g E G�, and any p, that APb9 is tempered and that e(AP69) = max {log i A. I I A. is an eigenvalue of AP(ii(g)) } o
Then
h(g) = J max e(AP1X9) = J max e(APf39) p
p
by earlier observations,
= J max e(APbp),by Proposition 94. 1 1 p
This completes the proof Theorem 94. 12 can in fact be extended to actions of lattices on manifolds. To see this, we first observe that the superrigidity theorem can be extended to actions of lattices by inducing. More precisely, we have the following 9.4.1 4 Theorem (A corollary of 5.2..5): Let r c G� be a lattice where G is a connected almost IF!-simple IF!-group with IF!-rank(G) � 2. Let S be an ergodic !-space with finite invariant measure. Let H be a connected IF!-slmple Lie group (trivial center) with HR not compact.. Suppose IX · S x r -> H11 is a cocycle with algebraic hull HR. Then there is an IF!-rational homomorphism n . G ---> H such that IX � 1Xni f . Proof: Let f : G�j[' x G� -> r be a strict cocycle corresponding to the identity map 1 -> r (4..2. 1 3). Define the G�-action induced from the r -action on S, X = G�/1 x 1 S (4. 2.21). Define a cocycle {J : X x G� -> HR by {J((y, s), g) = IX{s, f(y, g)).. Then X has a finite G�-invariant measure, and the algebraic hull of f3 is H11 . To see this last assertion, assume that f3 is equivalent to a cocycle with algebraic hull L11 c H11, i e , there is a {3-invariant function cp : X -> H11/L11 (4.2. 1 8(b)). Let r act on F(S, H11/ L11) by the a-twisted action. Under the identifi-
Ergodic theory and semisimple groups
1 82
cation F(X , H'R/L"') with F(G�jr, F(S, H'R/LrK)), cp corresponds to an f-invariant element of F(G�jr, F(S, H'R/L'R)).. By (4. 2. 1 9), there is a r-irivariant element in F(S H'R/ L'f 0 . That the same result is true for any local field of characteristic 0 is a result of Bruhat and Tits. See [Prasad 3] for a simple proof. ) Let G =
11 (Gp)CQp ' so that G is
peS
a locally compact group with the product topology. Lattices and actions of such groups arise naturally via "arithmetic" construc tions. For example, let us assume oo E S. Let Zs be the subring of CQ consisting of all rational numbers whose denominators have all prime factors lying in S. Clearly, if S # { oo }, Zs is dense in IR. However, we have a natural map Zs --+ CQ P for each p E S, and hence a natural map Zs --+ 11 COP · Th� image of Zs will then peS be discrete and in fact one can show that the image of Zs is a (cocompact) lattice in
11 CQ · More generally, suppose G is a connected algebraic CQ-group. peS P
We then have a natural embedding of Gz, --+ 11 Gcop ' whose image is discrete. peS In fact, we have:
10.1.1
Theorem
[Borel 6]: If G is a connected semisimple CQ-group, then Gzs is
a lattice in 11 Gcop peS This is the prototype of an "S-arithmetic" group. (See Definition 1 0. 1 . 1 1 .)
188
Ergodic theory and semisimple groups
Now let G be as in the first paragraph of this section. Any lattice in G is Zariski dense when projec;ted to each (Gp)rpp (This follows from Theorem 3..2 5 and the remarks following its proof.) A lattice r c G is called irreducible ifTH is dense in G for all H of the form (L)Q)p where L c Gv is an infinite normal proper algebraic (Qv-subgroup. Similarly, an ergodic measure class preserving action of G is called irreducible if the restriction to each such subgroup H is ergodic. Many of the results of the earlier chapters extend to this framework In fact, the proofs we have given generalize in large measure as well once one has developed some further algebraic background concerning the structure of algebraic groups over local fields. We shall therefore first indicate (without proof) one such basic algebraic result With this in hand, the extension of our earlier proofs to our current more general framework will then follow Suppose G is a connected semisimple k-group where k is a local field of characteristic 0. We define G + to be the subgroup of Gk generated by { Uk I U is the unipotent radical of some parabolic k-subgroup of G } . Then G + c Gk is a normal subgroup. (See [Tits 1 ] and [Bore!-Tits 2] for a full discussion of G + ) If Gk is compact (ie., k rank ( G ) = 0), then G + = {e}. On the other hand, for k = IR, and G almost IR-simple, if Gn:< is not compact then G + = G2 An analogous result in the case of an arbitrary k is given by the following theorem of Platonov verifying a conjecture of Kneser and Tits.. ..
-
10.1.2 Theorem [Platonov 1] (Kneser-Tits conjecture): Suppose k is a local field of characteristic 0 and G is a connected (algebraically) simply connected, almost k-simple k-group with k-rank (G) > 0.. Then G + = Gk If j:G --> H is a k-isogeny, then / ( G +) the following consequence.
=
H + Hence, Platonov's theorem has
1 0. 1.3 Corollary: Suppose k is a local field of characteristic 0 and G is a connected almost k-simple k-group with k-rank (G) > 0. Then [Gk:G + ] < oo.
The proof o f the vanishing o f matrix coefficients for unitary representations of connected simple Lie groups (Theorem 2 ..2..20), then yields the following result. 1 0. 1.4 Theorem [Howe-Moore 1] (Cj 2.2..20): With our assumptions as in 10 .. 1 .3, suppose n is a unitary representation of Gk so that niG + has no invariant vectors. Then the matrix coefficients of n vanish at oo in Gk .
This result enables us to establish analogues of all the versions of Moore's ergodicity theorem that appear in Chapter 2 .
1 89
Generalizations to p-adic groups and S-arithmetic groups
The following generalized versions of the stiperrigidity theorems (5. 1 .2, 5.. 2 .. 5 ) then follow with essentially the same proof With G = n (Gp)rop as above, we
peS
let rank (G)
=
L �p rank (Gp ) s
-
1 0.1 .5 Theorem [M argulis 1 ] : Suppose each GP is (algebraically) simply connected, (Gp)rop has no compact factors, and rank (G) � 2. Let r c G be an irreducible lattice. Let k be IR, IC, or �' for r prime and H a connected k-simple k-group. Suppose n:r --> Hk is a homomorphism with n (r) Zariski dense in H Then either (a) n(r) is compact (where the closure is in the H ausdorff topology); (b) For some p E S - { oo }, k = �p, and there exists a k--rational surjection qJ :Gp --> H such that the following diagram commutes. r
�r Hk
or c) k = IR or IC and there exists a k-rational surjection G oo --> H such that the corresponding diagram as in (b) commutes. 10.1.6 Theorem (Superrigidity for cocycles): Let G be as in Theorem 10. L5 and suppose X is an irreducible G-space with finite invariant measure. Let k, H be as in 1 0. 1 S. Suppose a:X x G --> H k is a cocycle that is not equivalent to a cocycle into a proper subgroup of Hk of the form Lk where L c H is a k-subgroup. Then either (a) a is equivalent to a cocycle taking values in a compact subgroup of Hk; (b) For some p E S - { oo }, k = �p, and there exist a k-rational surjection qJ:Gp -+ H and a cocycle fJ � a such that the following diagram commutes
or (c) k = IR or IC and there exist a k-rational surjection and a cocycle fJ � a such that the corresponding diagram as in (b) commutes. As in Chapter 5, from these results we deduce the following rigidity theorems.
190
Ergodic theory and semisimple groups
Theorem [Margulis 1 ] [Prasad 2] ( Cf 5. L L): Let S, S' be finite subsets of For p E S, let Gp be a connected, adjoint, semisimple {Qp group so that ( Gp ) 0. Let G
=
0 (Gp) 0 so m(n - 1 ( V) ) > 0.
n
Since n - 1 (U) :::> n - 1 ( V) n - 1 ( V), it suffices to show:
G is a compact set in G with m (A) > 0, then A - l A contains a neighborhood of e E G.
B.4 Lemma: If A
c
Proof: If Axn A -1=- 0, then x E A - 1A. So it suffices to show that {xE GJAxnA -1=- 0} contains a neighborhood of e. Since A is a compact set of positive measure, we can find an open set W => A such that m ( W) < 2m(A).. Again by compactness of A, we can find a symmetric neighborhood N of e such that Ax c W for all x E N. Since m (Ax) = m(A) and m ( W') < 2m(A), we must clearly have Ax n A #- 0 for
X E N. In 2.2. 1 6, we showed that an essentially invariant function on a G-space agrees a.e. with an invariant function. We now consider the analogous fact for maps between G-spaces.
Proposition: Suppose (X, J.l) is a standard Borel G-space with quasi-invariant measure and that Y is a standard Borel G-space. Supposef:X � Y is Borel and for each g EG,f (xg) = f(x)gfor almost all x. Then there is a G-invariant conull Borel subset X 0 c X and a Borel G-map rx 0 � y such that I = l a. e. B.5
Appendices
1 99
Let X0 = {x E X i g l-4f(xg) g - 1 is essentially constant in G } . For x E Xo, let J(x) E Y be such that f (xg)g - 1 = l (x) for a. e.. g E G. As a Borel space we can assume G c [0, 1] by A l, and hence by Fubini Xo is conull and ! is Borel (by an argument as in B. 2) . We have f = Ja e , and from the expressionf (xhg) g - 1 = [f (xhg) (hg) - 1 ] h, we deduce that for x E X0, h E G, we have xh E Xo and lcxh) = J(x) h Proof:
Suppose now that (X, fJ.), ( Y, v) are standard G-spaces with quasi-invariant measure . Let B (X, fJ.), B( Y, v) be the corresponding measure algebras. (Cf. 8.1). B.6 Corollary: Suppose cp:B( Y, v) --+ B(X, fJ.) is a continuous infective Boo lean homomorphism which is a G-map.. Then there is a conull G.-invariant Bore! set X o c X and a G-map J:x o --+ Y such that 1 * = cp. Proof: By general measure theory there is a Borel mapf:X --+ Since cp is a G-map, f will satisfy the hypotheses of B. 5.
Y such that [ * = cp.
B.7 Corollary: Suppose (X, fJ.), ( Y, v) as in B.6 and that cp is a bijection.. Then X and Y are conjugate G-spaces . le , there are conull, Bore!, G.-invariant sets X0 c X, Y0 c Y and a measure class preserving Bore! isomorphism } X 0 --+ Yo which is a G-map. Proof: Choose!as in B..6 . Since cp is surjective from general measure theory there is a conull Borel set X 1 c X o on which ! is injective . Let X 2 = {x E X 1 lxg EX 1 for almost all g E G } . By Fubini 's theorem, X2 is conull and Borel. We claim that lis injective on X2 G. Suppose x, y E X2 and J(xa) = Tc yb) for a, b EG . Since x, y E X 2, there is some g E G such that xag, ybg E X 1 . (In fact, this is true for almost all g ..) Since l is a G-map, we have J(x ag) = Tc ybg), and by injectivity ofl on X 1 , xag = ybg. Thus, xa = yb, establishing injectivity on X 2 G. By A.4, it clearly suffices to see that X 2 G contains a conull, G-invariant Borel set.. This follows from the generally useful:
Suppose X is a standard Bore! G-space with a quasi-invariant measure, and that A c X is Bore!.. Then there is a Borel subset B c A which is conull in A, and such that. (i) BG is Bore!; B.8
Lemma:
200
Ergodic theory and semisimple groups
(ii) There is a Borel map cp:BG -+ G such that for all x E BG, xcp (x) E B, and for all x E B, cp (x) = e. Proof: By 2. 1.1 9 and AA, we can assume that
X is compact metric and that G acts continuously on X. By AJO there is a Bore! set B c A which is conull in A and is a countable union of compact sets B; .. Since G is also a countable union of compact sets, BG will be as well, and in particular BG is BoreL To see (ii), it suffices to construct for each compact K c G and each i, a Borel map cp:B;K -+ K such that x · cp (x) E B; for all x E B;K. However, this follows from A.5 We now turn to the relationship between cocycles and strict cocycles, notions introduced in section 4.2. We recall that for transitive G-spaces, this relationship is clarified by 4..2J 5. Here we deal with the general case. Theorem: Let S be a standard Borel G-space with quasi-invariant measure Suppose H is a topological group whose Bore[ structure is countably generated (e.g. H second countable). Let et.:S x G -+ H be a cocycle. Then there is a strict cocycle [J:S x G -+ H such that for all g E G, [J(s, g) = et. (s, g) for almost all s E S. B.9
f.L
X = { (s, g) E S x G[h -+ et. (s, gh) et. (sg, h) - 1 is an essentially constant H-valued function of h E G } . For (s, g) E X, let & (s, g) be this constant. Then Fubini's theorem (recall H c [0, 1 ] as a Bore! space by AJ) shows that X is a conull Bore! set, &:X -+ H is Bore! and a = et. a. e. Suppose (s, g), (sg, a) E X. Then Proof: Let
the relation
et. (s, gah) et. (sga, h) - 1 = [ et. (s, gah) et. (sg, a h) - 1 J [et. (sg, ah) et. (sga, h) - 1 J implies that (s, ga) E X and a(s, g) &(sg, a) = &(s, g a). Now let So = {s E S[ for almost all g s G, (s, g) E X and (sg, g - 1 ) E X} . Since X is conull and Bore! in S x G and (s, g) -+ (sg, g - 1 ) is measure class preserving, it follows from Fubini's theorem that So is a conull Bore! subset of S. We now claim that if s E So and a E G with sa E S0 , then (s, a) E X. By the preceding paragraph it suffices to show that for some g E G, we have (s, ag), (sag, g - 1 ) E X. Since s E So, { g[(s, ag) E X} is conull in G, and since sa E So, { g[(sag, g - 1 ) E X} is conull in G. Therefore, our assertion follows. Summarizing we have produced a conull Borel set So c S such that if s, sg, sgh E So for g, h E G, we have &(s, gh) = a(s, g)&(sg, :h). By passing to a conull Bore! subset, we can also assume So satisfies the conditions Of B.8. Thus, S0 G is a conull Bore! G-invariant set and we can find a Bore! map .
201
Appendices
cp:S 0G -+ G such that cp(s) = e for sE So and s· cp(s)E S0 for all sES 0 G. We now define [3:S 0 G x G -+ H by [J(s, g) = & (scp (s), cp (s)- 1 gcp (sg)). From the conclusion of the preceding paragraph, it follows that [3 is a strict cocycle. Finally, for s t/= S0 G, define [J(s, g) = e for all g. Then [J:S x G -+ H is a strict cocycle and [3 = a a.e To complete the proof, let Go = { g E Gia(s, g) = [J(s, g) for almost all s} .. By Fubini, Go c G is conull and from the cocycle identity we deduce that G 0 is closed under multiplication. From B.l, it follows that Go = G. We now show that a continuous G action on B (X) actually defines a G action on points.
B.l O Theorem [Mackey 2]: Let (X, J1) be a standard Bore! space with a prob ability measure. Suppose G acts continuously on B(X) so as to preserve the Boolean operations. Then there is a standard Bore! G-space Y with quasi-invariant measure v, conull Bore! sets X 0 c X, Y0 c Y, and a measure class preserving Bore/ iso morphism cp:X 0 -+ Yo such that cp * :B( Y) -+ B (X) is a homeomorphic G-map.
a:X
x
G -+ X such that for each g E G the map ag:X -+ X given by ag(x) = a(x, g) is a measure class preserving map with a; equal to the original action of g - 1 on B (X). It follows that a is almost an Proof: By A. 1 2, there is a Borel map
action. I.e., (i) for g, h E G, a (x, g h) = a(a(x, g), h) for almost all x E X; (ii) for each g E G, a(a(x, g), g - 1 ) = x a.e. Choose an injective Borel function f:X -+ [0, 1]. Define h:X -+ Loo (Gh (the latter being the unit ball in L 00 (G), with the weak- * -topology) by (h(x))(g) = f (a(x, g)).. Let X 0 = {x E XIa(a(x, g), g - 1 ) = x for almost all g E G } . By (ii) and Fubini, Xo is a conull Borel subset of X. If x, y E X and h (x) = h (y), then for almost all g E G, a(x, g) = a(y, g).. Thus if x, y E X 0 , we deduce x = y, i.e.. h is injective on X 0 Then one checks that ( Y, v) = (L00 (G) l , h * (J1)) and Y0 = h (Xo) satisry the required conditions . (For a proof verifying all details, see [Ramsay 1 ] . )
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Subject Index adjoint group, 37 affine action, 60 affine space over S, 78 affine variety, 32 algebraic hull of a cocycle, 167 algebraically simply connected, 37 algebraic universal covering, .37 almost invariant vectors, 1 30 almost k-simple, .37 ex-invariant, 72 ex-twisted action, 65 amenable action, 78 amenable group, 59 arithmetic subgroup, 1 14 barycenter, 6 1 Bernoulli shift, 1 77 Boolean a-algebra, 1 50 Borel density theorem, 4 1 Borel map, 1 94 Borel space, 1 94
entropy, 1 76, 1 77 equivalent cocycles, 65 ergodic, 8 essentially constant, 1 1 essentially free, 68 essentially invariant, 2 1 essentially rational, 5 5 essentially transitive, 8 Gaussian, 1 10 induced action, 75 induced representation, 74 irreducible action, 20 irreducible lattice, 1 8 irreducible variety, .32 isogeny, .37 isomorphic actions, 8 Kakutani-Markov theorem, 59 k-almost algebraic, 40 Kazhdan property, 1 .30, 1 65 k-cocompact, 47 k-group, .3.3 Kneser-Tits conjecture, 1 88 k-rank, 8 5 k-simple, .37 k-split, 85 k-variety, 33
cocompact, 1 cocycle, 65 cocycle reduction lemma, 1 08 Comm (1), 1 22 commensurability subgroup, 1 22 commensurable, 3 connected algebraic group, 35 contracting automorphism, 1 52 convergence in measure, 49 countably generated, 1 0, 1 94 countably separated, 1 0, 1 94
lattice, 1 Lebesgue density theorem, 1 54 local field, .34
derivative cocycle, 67 direct integral, 23 Dye's theorem, 82
Mackey range, 77 matrix coefficient, 2.3 mean, 1 .3 .3
Su�ject index
measure algebra, 1 50, 1 96 measure class, 8 minimal action, 1 83 modular cocycle, 83 modular flow, 83 Moore's ergodicity theorem, 1 9, 21 multiplicative ergodic theorem, 1 78 orbit equivalence, 68 parabolic subgroup, 47 P(G), 1 3 3 Poincare recurrence, 1 65 properly ergodic, 8 property T, 1 30 quasi-invariant, 8 quasi-projective variety, 33 Radon-Nikodym cocycle, 67 rank, 85, 1 8 8 rational function, 32, 33 regular function, 32, 33 restriction of scalars, 1 1 6 IR-rank, 8 5
209
S-arithmetic, 1 9 1 skew product, 7 5 smooth, 1 0 stabilizer, 1 2 standard Borel space, 1 94 strict cocycle, 65 strictly equivalent, 65 superrigidity, 85, 98, 1 89 submatrix, 143 tempered cocycle, 178 twisted action, 65 UCB(G), 1 36 unipotent group, 52 unipotent representation, 53 variety, 33 von Neumann selection theorem, 196 weak containment, 143 Zariski topology, 32
