A. I. Maltsevs problem on operations on groups

A. I. MAL'TSEV'S PROBLEM ON OPERATIONS ON GROUPS UDC 512.543

A. Yu. Ol'shanskii

In response to A. I. Mal'tsev's problem (1948), associative operations are constructed on the class of all groups; these operations are distinct from the direct and free products and are hereditary for subgroups of the factors.

We say that an exact operation ~ is defined on the class of all groups if a group ~,, ~ J f ,

is assigned to each family of groups ~ = ~ ~

and monomorphisms ~ @ ,

are given

whose images (they are called the factors and are also denoted by ~,) generate the group ~, and all isomorphisms of the factors can be extended to isomorphisms of their ~ (i.e. the operation ~ is abstract). The classical examples of exact operations are the direct and free products of groups which also satisfy a number of additional postulates. These properties include: i.

Mal'tsev's Postulate.

inclusions ~$ ~

It says that for any subgroups ~'

can be extended to a monomorphism of the group

words, subgroups of the factors generate their ~ 2. the group

The Associativity Postulate.

unique isomorphism between ~

~

and

~o

~

into ~.

~,

the

In other

in

If a set Jr is partitioned into subsets J[~, ~.~,

~ is naturally isomorphic to the group

mented by the condition that if

of the factors

~o

~o ~.

then

This requirement is comple-

are trivial groups for ~ K ~ J K ,

then there exists a

~~

Main examples of assciative operations include the ~ -verbal multiplication which allows us to extend the free product inside an arbitrary variety of groups ~ to the class of all groups. Typical Mal'tsev operations are given by Grunberg--Shmel'kin multiplications. Axiomatic study of operations on groups has been initiated by O. N. Golovin in the 40s. The survey [i] deals with problems of classifying the operations. That article again raises, as one of its main open questions (Problem A), A. I. Mal'tsev's problem of 1948: do there exist associative Mal'tsev operations apart from the free and direct products? The authors of [i] view this question as "the central problem in the entire abstract theory of exact operations." It is also raised in the book [2] (p. 475). An approach to solving Mal'tsev's problem was found by S. I. Adyan [3] who noted that the classification of periodic words created in [4] and [5] can be naturally used here with modified rules of syntax (free product instead of a free group). The n-periodic multiplication introduced by S. I. Adyan (n is odd, n ~ 665) possesses a number of interesting and surprising properties; unfortunately, however, it can be defined only in the class of groups with no involutions [6] (and it is a Mal'tsev associative operation inside that class). The purpose of this article is constructing a Mal'tsev associative operation in the class of all groups, i.e., a solution to A. I. Mal'tsev's problem stated above. In the case of factors with no involutions and of a sufficiently large odd n, the constructed operation essentially coincides with S. I. Adyan's n-periodic product (however, the identification of these operations is a separate question because the scheme proposed in [7] substantially differs from the Novikov--Adyan approach even in its basic definitions). The operation defined

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 225-249, 1989. Original article submitted May 20, 1987. 2468

0090-4104/90/5104-2468512.50 9 1990 Plenum Publishing Corporation

below for (even or odd) n _> i0 l~ and arbitrary factors 9~,- is denoted by ~n like in [3], although for an even n or in a presence of involutions in the factors the group (~ need not he periodic even if the subgroups ~/~. are periodic. Elements of ~ that can be written as products of pairs of involutions may have infinite orders. We turn to an exact definition of the group tained from the free product ~ 0 = * ~

@=~ngl~, n ~

I0I~

The group ~

will be ob-

by imposing on it additional relations of the form

C n = i, where C is a word in the alphabet ~, obtained as a disjoint union of the sets ~\{|). For i > 0 we shall define periods of rank i and groups ~$~. To this end, we consider the set of all words C of length i in the alphabet ~ subject to the conditions: !) C is not conjugate in the rank i - 1 (i.e. in with an element of any ~;

@i-~) with any word of length i, i.e.

2) C has infinite order in the rank i - i; 3) C is not equal to a product of any two involutions in @~-~ and choose in it an arbitrary maximal subset $F~ of pairwise nonconjugate, in the rank i - i, words Ci, K such that no Ci, K is conjugate to Ci,~ z~ in the rank i - i for ~ # v. Next, put ~ = { C "~ ~

U ~

~, C ~ } U ~ - ~ ,

where ~ 0 = ~ ,

we define @ =(@~0; R = I , R ~ >

f=l

and @~=(~0; R =l, R ~ ) .

and put

Finally, for

~r ~t6J~

THEOREM. For each n -> i0 ~~ the operation defined above is an exact associative Mal'tsev operation on the class of all groups. The group

~=~n

is infinite if at least two factors are not trivial.

Each element

g e ~ is either conjugate in ~ to an element of one of the factors ~, or is conjugate to a power of some Ci, K (and, thus gn = i), or has infinite order and is equal to a product of two involutions. Periods of all ranks have order n in ~. i.

The proof substantially relies on [7] whose lemmas are stated in a modified form in Sec. The values of all parameters introduced in [7] are preserved.

i.

Properties of the Groups (~ We

begin

with necessary changes and modifications

in terminology.

Each word in the alphabet ~ can be uniquely written in the form WI..oWk, where all letters appearing in Ws lie in one ~(0, and u(s # ~(~ + i) for s = i, .... k - I. Here, Ws is a syllable of the word W. Similarly, if in some map (p. 317 in [8]) a letter ~ (e) in ~, is assigned to each edge e in such a way that ~(e -l) =~(e) -I, then each path p in this map is divided into segments according to the division of the word ~(p) into syllables. The number of syllables of a word W (the number of segments of a path p) is called the IWI of the word W (the length IPl of the path p). The notation U ~ V is viewed in this article as the syllable-wise equality of the words U and V (instead of the graphical equality in [7]). An irreducible word is one in which each syllable al...a s possesses the property that ak...a ~ # 1 in the corresponding group ~ for 1 ~ k 5 s i s. If, in addition, the first and the last syllable of a word W lie in different ~, ~ (in the case of IWI > i), then W is said to be a cyclicaily irreducible word. A c_.vclic shift of a word W is defined as any word W' such that for some s we have W' ~ UW~+I...WkW~...Ws where VU = Ws in ~t). A subword in an irreducible word W is any irreducible word with a division into syllables of the form UWk...W~V, where k S s + I, and the syllables U and V lie in the same factors as the syllables Wk_ I and Ws respectively. A word C is said to be si__~le in the rank i ~ 0 if: i) C is not conjugate in the rank i to a power of any period of rank k 5 i; 2) C is not conjugate in the rank i to a power of any word of length less than ICI;

3) Icl e 2. 2469

The definition implies that a word simple in a rank i is simple in the ranks i. The periods of rank i + 1 are simple words in the ranks 5i. Proof. Statements i) and 2) obviously follow from the definition of periods of ranks i, 2, ... and the group ~, because in a dihedral group each element is a product of two involutions or is conjugate to one of the two generating involutions. In the case of 3), we choose a word D according to Lemma 1.5, IDI > i. Like in Le~ma 5.6 in [7], we deduce that D has infinite order in the rank i because [n/6] > 8. The word D is also not conjugate in the rank i to an element of ~, because considering a minimal ring diagram of such a conjugacy we arrive at a contradiction to Lemma 5.5a since [n/3] > 8. i if D = IJ is a product of two involutions in the rank i, then D and D -I are i conjugate in the rank i, D -I = IDI -~. Let A be a minimal ring diagram of rank i for conjugacy of the words D and D -I. By Lemma 5.5a, A has no cells because [n/3] > 8. Thus, D and D -I are conjugate in G0. But then the cyclic irreducibility of the word D would imply that D -l is a cyclic shift of the word D and, therefore, D has subwords S and S -l, where ISI 1/21D I, contrary to the choice of the word D. Finally,

Thus, if there are no periods of ranks i + 1 ..... ID[ - i, i.e.,~i = J ~ + l .... =~lOl--1, then, by definition, D is conjugate in the rank i to some period of the rank IDI > i.

2471

Suppose now that some period C of the rank i + i is not simple in the rank i. By the definition of a period and by Lemma 4.6a, the word C is conjugate in the rank i to a power of some word C simple in the rank i, where ICI < ICI, i.e., j = ICI 5 i. But then C, like C, has infinite order both in the rank i and in the rank j - I. Thus, the definition of periods of rank j implies that C is a product of two involutions in @~_~. In this case, C is a product of two involutions in the rank i because a power of a product of two involutions is also a product of a pair of involutions. The obtained contradiction to the definition of a period of rank i + i shows that C is simple in the rank i and, thus, also in every rank k 5 i. The statement of Lemma 6.1a is as before. It appears that a new possibility arises in part 3) of the proof: D is conjugate in the rank i to an element C of ~ (the other two possibilities, i.e,, that D is conjugate in the rank i to a power of a word C simple in the rank i or to a power of some Ck, ~ have been considered in [7]). We will show that in fact it cannot arise. Indeed, like in part 6 of Lemma 6.1 in [7], in this case we obtain the equality i YZIX = Z2, w h e r e , ( Y ) is formed as a product of the labels of edges of a path p. Next, like in [7],

IZ~! + Iz~l< ~(p-'--l)lq~l +--~- !A[, and X is an A-periodic word,

IXI >- lq21 > k11Al"

Thus,

P-~ + --~- I A I >~(p-,--1 )lq21 +~-~ IAI+~>IZ~I+IYI+ IZ~l, p-~ iXl>--c-Iq~l which contradicts Lemma 4.2a applied to a minimal, i X = ZI-IY-IZ2 -I

in the rank i, diagram of the equality

Both the statement and the proof of Lemma 6.2a remain unchanged. LEMMA 6.3a. Let & be a minimal diagram of rank i with a contour PlqlP2q2, where ~(ql) a n d S ( q 2 ) are periodic words with a period A simple in the rank i and max(Ip~l , IP21) < ~IAI Then max(lqll , lq21) ! kslA I or A is a product of two involutions in

~{.

Proof.

If, say, lqll > kslAl, then using the notation of Lemma 6.3 in [7], we obtain i i the equality Z 2 = i as it was obtained there. Furthermore, the equality A = Z-IA-IZ implies i that (AZ) 2 = I (here, Z and AZ are not trivial in the rank i because, by Lemma 4.6a, a word A simple in the rank i has infinite order in ~{.) So A = (AZ)Z is a product of two involutions in ~{. 9 LEMMA 6.4a. Let A be a minimal diagram of rank i with a contour PlqlP2q2, where ~ (ql) and ~(q2) are periodic words with periods A and C simple in the rank i, lq21 ~ ~nlCl and max(let1,

IP21) < ~nb, where b = min(IAl,

ICl).

Then either

lqll < (i + e)IAl, or A is con-

jugate to C • in the rank i. In the latter case, if A m C • then either A m C -I and ql, q2 are compatible in A or A is a product of two involutions in ~ r The proof changes only at the moment when Lemma 6.3a is applied; a change in its statement has caused the corresponding modification in the statement of Lemma 6.4a relative to [7]. 2.

Mal'tsev's Postulate for n-Products

LEMMA i. In all groups ~i, and thus also in 6, each element of ~ is distinct from i, i.e., ~ is generated by isomorphic copies of the groups ~ , ~ J K . Proof.

The statement follows from Lemma 4.3a and the inequality on > I.

Henceforth, until the end of Sec. 2, ~='

are subgroups of the groups

~, ~'=U~'\{|}.

LEMMA 2. If all segments of a contour p of a minimal diagram A of rank i have labels in ~', then the segments of all cells of the diagram A also possess this property. Proof. Arguing "by contradiction," we choose a counter example A with a minimal number of cells. By Lemma i, IPl > i and the word ~ (p) may be assumed cyclically irreducible. By Lemma 5.5a, A has a cell ~ with a contour PlP2, where Pl is a subpath of p and Ipll > 1/318HI.

2472

Since n > 4, the labels of all segments H out of A and replacing the subpath Pl and a similar condition of the segments statement of the leamma, A does not have

of the cell ~ lie in ~'. Thus, excising the cell in p by p2 -I, we obtain a diagram A' with fewer cells of a contour. Since A' has no cells refuting the them either, a contradiction, a

LEMMA 3. Let q be a smooth contour of a minimal ring diagram A of rank i with contours q and t. Then if letters of ~ occur in ~(q), then they also occur in ~(t). Proof. The statement is obvious there exists a cell H whose incidence like in Lemma 5.5a, results in a cell and p is a cyclic subpath in t. Thus, possible.

for r(A) = 0. Otherwise, by Lemmas 2.3a and 3o6a, degree for t is greater than 8 - ~ > 1/3 + 400u which, ~ with a contour of the form pp, where IPl > 1/3 18~I like in the proof of Lemma 2, an induction on T(A) is

LEMMA 4. If words C -l and C are conjugate in rank i, then C is conjugate to an element of some factor ~, or C is conjugate to a power of some period of rank k 5 i, or C is a product of two involutions in U i Proof.

If neither of the first two possibilities occurs, then, by Lemma 4.6a~ the word

C is conjugate to a power of some word D simple in the rank i, i.e., D E ~ XD-~X -I, ~ > 0, for i some X. But then D s~ = XD-S~X-I. Applying Lemma 5.3a to the diagram of this conjugacy for a sufficiently large s, we excise out of it a subdiagram to which we already can apply Lemma 6.3a. This implies that D is a product of two involutions in ~i, because a power of a pair of involutions is again a product of two involutions. We now denote by

~i' the group obtained by imposing on C 0 " = , ~ /

relations of ranks

i, ..., i defined in the alphabet ~ in the same manner as relations of rank i have been defined in the alphabet ~. In order to avoid a confusion between relations of the two sorts, in the following lemma, while considering the group @/, we will talk on its relations of ranks i', 2', ..., i' in the alphabet ~. By Lem~a I, in the rank i' the group ~ ' is generated by subgroups isomorphic to the groups ~/. On the other hand, ~ / may also be viewed as subgroups of ~i, generating some subgroup @i in ~. LEMMA 5.

There exists a natural isomorphism between the groups

@/

and

~i

Proof. We perform an induction on i with an obvious base of i = 0. First, we will prove that each relation of the form Ci,K n = 1 of the group ~ / also holds in @~. Suppose first that a word Ci, ~ is conjugate in the rank i - 1 to an element of ~, and consider a minimal ring diagram A of rank i - 1 for this conjugacy with contours p and q, where ~(p)~---Ci,• ~(q)~. Applying Lemma 5.5a to A, we find a cell ~ in A with a contour PlP2, where Pl is a subpath of p and IPll > 1/313HI" Since n > 4, the labels of all segments of the contour of the cell H, like the labels of segment of the path p, lie in ~'. Since r(~) < i, by the induction hypothesis, the relation in ~ , corresponding to the cell H also holds in @'i-i, i.e., when the subpath Pl in p is replaced by p -i, we obtain a path p such that ~(p) m Ci,, 1 if iICI] > I. Unlike the length ICj El, the norm lICj (n~ - 4)IICj'II, and IIZ~II, JlZ211 < 51jCj'li. Hence Y = W, where

< (46yn + IO)IICj'II, contrary to Lemma 12 applied to the diagram of this equality because

9 (n[~-4) >46yn+lO. Thus, ~

is also a large cell9

By Lemma 2.2a, j < ~ / 2 k .

(1) By Lemma 19,

IICj'l] < 22u

and, by Lemma 18, the label of a subpath of the contour of the cell ~ is equal in the rank j - 1 to a word U, where

Hence

fl X~ [[ < 22y (n + 10)II C~ II + 23yn II Cj J] < (22y (n + 10) -~ 22y. 23yn) [j C~ J] < 23yn [I C~ [tLEMMA 7. Let A be a o-regular minimal (ring) diagram of rank i, F the subdiagram of incidence of a large cell ~ with a period C k to a (cyclic) o-smooth region with a period C', where C is simple in the rank i or C = - C s + i , ~ < i. Then, if the degree of F-incidence of the cell ~ is no less than e - ~, then IJF A qll < (! + e)IJC'II. Proof. Since A is o-regular, j = r(F) < k. By Lemma 6, one can apply Lenmla 21 in the rank j < i to r, and if the conclusion of Lemma 7 were false, then Ck -I and C' would be conjugate in the rank k - 1 and the cell ~ would be compatible with q, contrary to the c-smoothness of the region q. LEMMA 8. Let A be a o-regular minimal (ring) diagram of rank i, F the subdiagram of incidence of degree ~8 - ~ of some cell ~ of rank k in A to a c-smooth (cyclic) region q with a period C~'. Then k < I. Proof. q~ = F A q.

First, assume that ~ is a large cell. Let PlqlP2q2 be the contour for F, where Like in the proof of Lemma 6, we conclude that C~ is a large period [inequality Z-1 (i) is preserved when ~ is replaced by e - a). By Lemmas 7 and 18, ~(q2) = V, where IV! < IoJc~J. The last equality also holds in the rank k - 1 if s ~ k. By Lemmas 2.1a and 4.2a, this implies that p(e - a)nJCkJ < IOJCs + 4ynICkJ, and thus [CkJ < JC~I , i.e., k < s If ~ is a small cell, then, by Lemma 6 for Pl and P2 and by Lemma 12 in the rank j, ;lq2;l < p-llfplqlp=ll = p-l, i.e., llq2H = i. Assuming that Cs is a large period, we deduce that k < E as before. If Cs is a small period, then q is a smooth region, and k < E by Lemma 2.3a. i LEMMA 9. In a o-regular (ring) diagram of rank i the degree of incidence @ of each cell ~ to a c-smooth (cyclic) region q of the contour is less than ~. Proof. Let F be the correspondent subdiagram of incidence with a contour PiqlPzq2, where ql = F A ~, q2 = F A q. By the hypothesis, j = r(F) < k = r(~). Let C' be a period for q, where C is simple in the rank i or a period of rank E > k by Lemma 8. If ~ is a small cell, then, like in Lemma 8, Jlq2iJ = i. Assuming that ~ > ~, we arrive at a contradiction like in the proof of Lemma 2.3a (to the minimality of o-syllables of the word C' in the rank j < s Henceforth we assume that ~ is a large cell. 2475

By Lemma 6, the w o r d s ~ (p~) and ~(p=) are equal in the rank k - 1 to words x~ and x2 k-z such that llx~ll, fix211 < 23~nllCk'll and, by Lemma 18 in the rank k - i < i , ~ ( q ~ ) = Z~YZ=, where Y is a Ck-periodic word, [IYII > (na - 2)llCk'll, IIZ~II, llZ211 < 511Ck'll. Since C is simple in the rank k - i, one can apply Lemma 12 to a region with the label Y in a minimal diagram k-~ A0 of the equality Y = ( Z = X ~ ( q 2 ) X ~ Z z ) -~ and p ( a n - - 2 ) l I C / [ l
n ( p ~ -- 47~)llCk'll.

On the other hand, by Lemma 18 for a path complementing q~ to a contour of the cell ~, k we conclude that ~(q2) = W, where IIWIJ < (46~n + i0 + (i - a)n)llCk'll < (i - a + 47~)nJICk'JJ. Now, if lJq2Jl 5 IIC'II, then we have a contradiction like in Lemma 2.3a because

0a-47?> 1-~+47y.

(2)

If Jlq211 > llC'll, then, again like in the proof of Lemma 2.3a, we get a contradiction (by means of Lemma 7 instead of Lemma 2.2a) because

(p~-47y) (l-e)> Lemmas 10-14 a r e p r o v e d by a c o l l e c t i v e

(I-a+47?) (I+e)+2n-L induction

on r ( A ) .

LEMMA 10. L e t F be t h e s u b d i a g r a m o f i n c i d e n c e o f a l a r g e c e i l ~ w i t h a p e r i o d Ck t o a ( c y c l i c ) r e g i o n q o f t h e c o n t o u r o f a m i n i m a l ( r i n g ) d i a g r a m A w i t h an i n c i d e n c e d e g r e e ~, l e t PxqlP2q2 be i t s c o n t o u r , w h e r e q l = F A ~, q2 = F A q, and q c o m p l e m e n t s q l t o c o n t o u r o f the cell ~.

Then

I[ q~ II~ (p~--47y) n l[ C'~ II, a ~

-

1

k--I

(p~q- P2) = W ,

r~e Jl W ]]
e. By Lemma i0,

Ilhil> (p(e-a)-47v)nllC/ll.

(5)

By Lemma 6, one can r e p l a c e t h e p a t h s p~ and P2 in t h e subdiagram rz by pz, P2, where IIps

< 23~nllCj'll,

and

(pg)

j - =~

~(p~),

s = i, 2.

For the obtained diagram F z' we have

9 (Fz') < ~(A) since the cell ~ of rank j has been excluded from A. So, by statement b) of Lemma 16 (and Lemma 18 in the rank j - i), for each subpath ql of the path qz there exists a i word U~ such that ~(q~) = U~ and

II G 11< (2. 162yn + 10 + ~ -1

n) II C}]l-

(6)

In the diagrams A I and A 2 with the contours q'pz-Zws2-1t ' and q"t"sl-Zup2 -I, using Lemmas 6 and 18, one can replace the subpaths of the contours p1-Zws2 -i and sz-lup2 -l by u' and u", respectively, preserving the labels in the rank j - i, and

II u ' Ii + !1 u" II < (92u + 20 + (1 - - O) n) JJC} lJ. Applying Lemma 16(a) to the obtained diagrams

At' and A 2' [their types are less than t(A)], i we obtain for any subpaths ql and q2 of q' and q" words Vz and V 2 such that ~(q~) = V~, ~ = i, 2, and

!1V, II + li v, II < + I(93~, + l--e) n I! C} II + IIt' II + II t" I1]By (5), (6), and (7), for the subpath q there exists a word U such t h a t , ( q )

IIUll(~ ~2(sli)-Zw1(si~)-~1)

in the rank j - I), we conclude by Lemma 16(a) that for each subpath i of the path ql there exists a word U such that ~(q) = U and

Hence

ISI < (23 + 139)ynlC'l

= 162~nlC'I.

LEMMA 17. Let A be a minimal ring diagram of rank i with contours t and q, where ~(q) C is a word simple in the rank i and ~(t) m C'. Then: a) for each (cyclic) subpath ~ of the path ql there exists a word U such that ~(q) =i ilUl < 4/3 IC'I + 1/2. There exists a path s with no self-intersections in A connecting i points on q and t such that ~(s) = S, ISI < max(0.021tl; i);

U,

b) for each subpath t of the path t there exists a word U such that q (~) =i u, juJ < ICI + 1/2. There exists a path s with no self-intersections connecting t and q such that i ~(s) = S, ISI < 0.021q I. 4/3

Proof. The first half of statement a follows from Lemma 16a. To prove its second half, we select a e-cell ~ with incidence degrees ~1 and ~2 like in the proof of Lemma 16(a) (using the same notation). If ~ is a small cell, then ISI 5 1 by Lemma 6. If ~ is large, then, by

2481

Lemmas 6 and 18, one of the paths s2w-lpl,

s1-1up2 -I has a label equal in the rank i to a

word S, where ilSU < (467 + l--8/2)nliCj'll. [cf. (5)]. Thus, IISli < 0.02iltli.

On the other hand,

lltll ~ (p(8 - ~) - 477)nllCj'il

b. The first half of the statement follows from Lemma 15 (a). Its second half is obtained like in part a (using Lemmas 2.1a and 4.2a instead of Lemma I0). LEMMA 18.

a.

Let X be a periodic word with a large period C simple in the rank i, i ~IC1 ~ txi < (s + I)IC I for some s There exist words Z l, Y, Z 2 such that X = ZIYZ 2, Y is a C'-periodic word, ilZ111, UZ2U < 511C'll, (s - 2)ilC'll ~ liYii ~ s b.

Let X be a C'-periodic word, where C is a large period simple in the rank i, s 5 i llXil < (s + l)llC'li. Then there exist words ZI, Y, Z 2 such that X = ZIYZ2, Y is a C-periodic word, (s - 2)Ic I 5 IYi ~ Pica, and IZ~[, Iz=i < 41c i . Proof.

Let X ~ xIckx2, where s - 2 < k 5 s and XI and X= are the end and the bei ginning of the word C. By Lemma 17, C = ZC'Z -I, where llZll < 1/2 + 4/3 ilC'li + max (ilC'li/50; i I), and X d = Ud, d = I, 2, where ilUdll < 1/2 + 4/3 liC'll + i. This implies that it suffices to put ZI m UIZ, Z2 ~ Z-IU2. b.

a.

This is proved like part a, except that we use Lemma 17(b).

LEMMA 19. Let A be a minimal diagram of rank i with a contour PlqlP=q2, where ~ (ql) and ~ (q2) are periodic words with periods C and D, simple in the rank i, such that C is a i large period. Furthermore, let IP~I, IP=I < 27nICl and for some words X~, ~ = i, 2, ~ (ps = Xs and lIXs < 237niiC'II. Then, if lq~l ~ ~niCl, then either IIC'II < 227llD'II, or C and D • are conjugate in the rank i. In the latter case, if C m D • then either C m D -~ and q~, q= are C-compatible in A or C is a product of two involutions in ~ i. Proof. It is easily seen that relaxing the estimate for IPs in half in the proof of Lemma 6.4a results in a similar statement with e replaced by 2e. Therefore, we may assume i that lq=i < (i + 2e)ID I. It follows from Lemma 18(a) that ~(q=) = W, where ilWil < lliID'il. i Taking into account the hypotheses of the lemma, we have ~ (q~) = V, liVil < llllD'il + 467nliC'II. i = Z~YZ=, where lIZ~li, iIZ2il < 511C'll, and Y is a i C'-periodic word with ilYii > (n~ - 2)ilC'll. Since ~(q~) = V, V is equal in the rank i to some word U, where iIUil < llllD'li + 477nlIC'll. We apply Lemma 12 to a minimal, in the rank i, diagram of this equality: On the other hand, by Lemma 18, ~ ( q ~ )

p(n~--2)llC'll Sn[cl by [q=[ > (8 - ~)n[C[, and the conditions [pz[, [p~[ < 27n[C[ by [z~pz[, Ip~z~[ < 107yn[C[. Like there, we deduce that either [q'[ < (i + e/2)[A0[, or C • and rank j. In the latter case we may choose A0 m C +z-. Then, like in a product of two involutions in ~, or A 0 m C -m and the paths ql -~ in A 0. In the latter case A' and C -~ are conjugate in the rank i, C -~ and q~ and q= are A'-compatible in A.

A 0 are conjugate in the Lemma 6.4a, either C is and q' are C-compatible and if A m C • then A

It remains to consider the case of [q'[ < (i + e/2)[A0[. Suppose that llq21l~ (l+e)llA'll. We choose a vertex u on the path qz in such a way that it would divide the path q= = q= 1 q= 2 , 0

where ~ ( q i) m ~, is a cyclic shift of the word A' or, perhaps, ~(q=~) = A'a' for some letter a' e ~ (if q= has no secondary vertex determining a cyclic shift of the word A'). According to Lemma 20, we choose vertices os os z on the path qz and vs vs on the path q= in such a way that the point u lies on the subpath t" = vs - vs For the path p consisting of the edges of the paths t" and q22 we have [[pll ~ ~(i + g-l)llqa]I. Thus, if W ~ ~(p), where p = o0 - os then, by Lemma 12 for the subdiagram with the contour p~pts

II W II > ~ (1 + ~)-x II q~ ll--46vn II C' IIThis inequality also holds if C' is replaced by C, a duplicate for C in the rank j, because []~I[ ~ llC'll. Let [q~[ = mlC [.

By Lemma 18, ~(ql) j= Z~XZ2, where X is a C-periodic word,

HXII > (m - 2)IICll; IIZIII, llZ2ll < 5U~U. (pm - 47xn)lldll. Thus,

Comparing X with ~(q2),

IIWIl>p2e(I § If tradicts

by Lemma 12, we h a v e

IIq2ll >

9

0.8 mlcI, then, b y Lemma 18, ~ ( p ) ~ W, w h e r e the estimate (9) since m ~ (8 - ~)n. Therefore,

(9)

tIWll < ( 0 . 8 e m + 10)IICII w h i c h c o n -

Ipl > 0.8eml C I. We d e n o t e

t'

= os - os

and consider

(10)

a subdiagram

in A with

the

contour

t~-~t'ts

''.

By Lemma 20, [t~-It'ts < 20xn[C[. Since the region t" is A'-smooth, by Lemma 15(a) for the subpath s = vs - u there exists a word S such that

S =ir Let u'

be a vertex

of the

I S I < T4 .20ynICI path

q' which divides

+}!- PIP[ - [ z 2 P l [ - [R[ > ( 0 . 8 p e m - 2 4 2 y n ) [ C [ the hypotheses o f t h e lemma, t h e d e f i n i t i o n o f Z2, a n d i n e q u a l i t y (10). So

IAol > (e/2)-11 qg I > (l.6pm--60 000yn) I C IOn the other hand, by Lemma 4.2 in the rank j for A 0

IAo 1 6 0 250y. Thus, the inequality llq211 ~ (i + ~)lIA'll is false. 2484

The lemma is proved.

by

5.

The Proof of the Theorem

The exactness of the n-product has been established in Lemma 1 and Mal'tsev's postulate in Lemma 5. Adding trivial factors does not change the group ~, as follows from its definition. To prove the associativity,

we compare

@ with

@~=

Rn

@~, where

G~

~

Rn

~.

By the

above, both groups are generated by the subgroups ~, and we have to prove that the sets of relations between elements of distinct subgroups 9~g in G and in @o coincide. An irreducible word in the alphabet

9~ can be rewritten

in the alphabet

~=

U ~n \ {I}, g

when an element of some ~ is assigned to each o-syllable (as can be seen from Mal'tsev's postulate and the postulate on trivial factors, the subgroups ~,, ~ J f ~ , generate in ~ a subgroup naturally isomorphic to the subgroup ~ in ~o ). Turning from a word X as an element of the group @ to the same word as an element of @~, we mean rewriting of this kind; then IIXII = IX[o, the length relative to H. The reverse passage from ~ to ~[ is also possible, of course. As follows from the definition of duplicates of periods, the system of all relations of the form (Ci,K') n = 1 is defining for ~. Therefore, it suffices to prove, by collective I,K 'II and [Cj, K Io, that each defining relation (Ci,~a)n = i of the group induction on IIC" holds in Go, and all relations Cj,v n = 1 for @= are true in G. Let C i' be a duplicate of a period of rank i and suppose that (Ci')n # 1 in Goo Then it follows from the definition of periods and from Lemma 4.6a that the word C i' is conjugate in ~ to an element of one of the groups @n or is a product of two involutions. In the former case C i' = XaX -l in ~o, a ~ ~, and in the minimal diagram (over ~ ) of this conjugacy the lengths of perimeters of all cells are, by Lemmas 5.1a and 4.3a, less than Go

(i+2~) P-' ([I C/ll +1) a, then IIF A q'll ~ 3 by Lemma 6. Thus, excising ~ together with F out of A', we lower llpll + llqll by at least two units. By the induction hypothesis, one can join the contours in the obtained diagram A 0 by a path t o , i where ~(t0) = To, liT011 < llpll + l[qll - 2. Using Lemma 6, we can extend the path t o to q' while liT011 is increased no more than by i. This implies the assertion of the lemma.

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Suppose, further, that ~ is a large cell with a period Ck whose degree of F-incidence to q' is >8. If t = ~ A F, p~tp2s is the contour for F, and t is an arc of the cell ~ comi plementary to t, then replacing s by a path t, where 9 (s) = ~(t), according to Lemma I0, results in a diagram A, T'(A) < T'(A) (because the cell ~ has been excised), for whose contour q we have 11qil < ilqU + (i - 8 - pe + 937)niICk'li. Applying Lemma 22 to A, we find a path i _ joining the contours, where ~(t) = T, iITII < liqi; + ilpli + (i - 6 - p8 + 937)niICk'll. Adding to it a subpath of the path q with norm 5(i - 8 + 477)nlICk'[l (see Lemma i0) along with a subpath of norm !l (in the passage from A' to A by means of Lemma 2), we obtain the required inequality for UTII. Finally, let ~ be a large 8-cell incident to both p' and q'. and 12,

Then, by Lemmas 6, 18,

IIp'll+llq'[[ > (p0+93u

(12)

i and f o r t h e p a t h t ' = o l -- 02 ( s e e t h e n o t a t i o n i n p a r t 2 o f Lemma 5 . 1 a ) ~ ( t ' ) = T ' , where IIT'II < (1 - 0 + 47~)ntlCk'll by Lemmas 6 and 18, w h i c h t o g e t h e r w i t h ( 1 2 ) i m p i i e s t h e r e q u i r e d i n e q u a l i t y f o r t h e word T w h i c h i s o b t a i n e d by e x t e n d i n g t h e word T' by s u b w o r d s o f norm ~1 i n t h e p a s s a g e t o A. The 1emma i s p r o v e d ; t o c o m p l e t e t h e p r o o f o f t h e t h e o r e m , we assume t h a t where Ci i s a p e r i o d o f r a n k i f o r ~o. A g a i n , two p o s s i b i l i t i e s may a r i s e .

Ci n ~ 1 i n

~,

1. Ci i s c o n j u g a t e i n ~ t o an e l e m e n t a ~ ~,. By Lemma 22, Ci = XaX - z i n @, w h e r e fIX < 2(llCill + i). By Lemma 13, the last equality holds by virtue of the conditions with periods Cj, where [iCjii < 8p-~n-lilCili = 8p-~n-~ICil o. If all (Cj') n = 1 in ~, then C i = XaX -i in ~, whence a n = 1 in ~ (since Ci n = 1 in ~ ), i.e., an = 1 in ~ (since a ~ ~,~ @ ). Hence c i n = (XaX-i) n = 1 in @, contrary to the assumption above.

2. C i = XCi-IX -I in ~. By Lemma 22, we may assume that llXi[ < 411Ciil and, by Lemma 13, the given equality holds by virtue of the conditions with periods Cj, where [iCj'il < 10p -~ x n-lllCiil = 10p-ln-llCilo. If all (Cj') n = 1 in ~o, then C i = XCi X-~ in ~o, which leads to a contradiction (as a similar equality in ~ above). Thus, there exists a period Cj for the group @, such that ilCj'li < aCiDo and (Cj') n # 1 in ~o. Together with the conclusion drawn before Lemma 22, this means that the assumptions are false, and the reserves of relations for ~ and @o are identical. The remaining assertions of the theorem follow from Lemmas 4.6a and 5.6a. LITERATURE CITED i. 2. 3. 4. 5. 6. 7. 8. 9.

O. N. Golovin and M. A. Bronshtein, "Axiomatic Classification of Exact Operations," in: Algebra and Logic [in Russian], Nauka, Novosibirsk (1973), pp. 40-96. A. G. Kurosh, Group Theory [in Russian], Nauka, Moscow (1967). S. I. Adyan, "Periodic products of groups," Trudy MIAN, 142, 3-21 (1976). P. S. Novikov, "On periodic groups," Dokl. Akad. Nauk SSSR, 127, No. 4, 749-752 (1959). P. S. Novikov and S. I. Adyan, "On infinite periodic groups," Izv. Akad. Nauk SSSR, Ser. Mat., 32, Nos. i, 2, 3, 212-244, 251-254, 709-731 (1968). S. I. Adyan, "On simplicity of periodic products of groups," Dokl. Akad. Nauk SSSR, 241, No. 4, 745-748 (1978). A. Yu. Ol'shanskii, "On the Novikov-Adyan theorem," Mat. Sb., 118 (160), No. 2, 203-235 (1982). R. Lyndon and P. Schupp, Combinational Group Theory [Russian translation], Mir, Moscow (1980). S. I. Adyan, Burnside's Problem and Identities in Groups [in Russian], Nauka, Moscow

(1975).

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