Dimensional orbits in flat projective planes

Math. Z. 122, 117-124 (1971) 9 by Springer-Verlag 1971

1-Dimensional Orbits in Flat Projective Planes* HANSJOACHIM GROH

1. Introduction In [8, Lemma 10, 11], Salzmann showed that the 1-dimensional orbit of the automorphism group a P of a "hyperbolic" projective plane P is an oval by using the fact that a P ~ PSL 2 (R). The purpose of this paper is to investigate for arbitrary fiat projective planes P orbits of 1-dimensional subgroups A of aP. Our main result (Theorem 3) is that such orbits, unless they are contained in a line, are always ovals if A ~ S1 (Circle Group), including the above result. If A mR (reals), the situation is more complicated, but the orbits are either "spirals" or "nearly" ovals (compare also [1, 2] for the Desarguesian case). These results will be an essential tool in the classification of fiat projective planes with dim a P = 2.

2. Topological Properties of 1-Dimensional Orbits Throughout this paper, let (P, L) be a fiat projective plane, i.e. a topological projective plane ([5]) whose point space P is a 2-manifold. The only 2-manifold which is possible is the space P2 of the classical projective plane, and the lines must be Jordan curves, i.e. homeomorphic images of $1 ([-6]). For simplicity, we will often denote the plane by its point space P. The group: a P of all automorphisms, given the compact open topology, is a locally euclidean group of dimension < 8 and acts as topological transformation group on P and L ([,7, Satz 4.1 and Satz 3.1]).

Definition. An orbit of a subgroup A of a P is called trivial if it is contained in a line. Notations. For p~P, the set of lines containing p is denoted by p,. "-~" means isomorphic (e.g. for topological groups), but "homeomorphic" is abbreviated by ~. We write G < H iff G is a subgroup of H. If I is a subspace of X ~ R and a, b~l, ]a, b [~ stands for the intersection of the open interval { x ~ X ] a < x < b } with I. Similarly [-a, b[,l= {a} u ] a, b I-z, etc.

* Research supported by the National Research Councilof Canada, Grant A7336.

118

H.Groh:

L e m m a l I. If a P > A ~ S 1 o1" R, then A fixes a point p and a line I. For A ~-$1, p and l are uniquely determined, pq~l, and A is transitive on I. For A g R , precisely one of these cases holds: (1) p and I are uniquely determined, p$1. The group Z of(p, l)-dilatations in d is infinite cyclic, d is transitive on I. (2) Every fixed point of A lies on a fixed line.

Proof For the first statement, let flEA be an element of infinite order. By the divisibility of A, we can find recursively a sequence {fn} of elements satisfying f~+l=fn. All f, have again infinite order, and in particular the subgroup G generated by them is not cyclic and therefore dense in A. The set F(fn+ 1) of fixed points of f , + l is contained in F(f,). Also F(f,)4= ~, as every homeomorphism of P2 has a fixed point. Therefore, as the F(f,) are closed and have the finite intersection property, the compactness of P implies c~ F(f,) 4: 0, which implies a universal fixed point for G and therefore for G = A. If A -_ S 1, p a fixed point, I a fixed line, then the classification of all transitive, locally compact, connected, effective transformation groups on 1-manifolds by Brouwer ([3]; see [9, Hilfssatz 1.9] for modern terminology and proof), in the following quoted as Brouwer's theorem, implies that A is transitive on p . ~ S t . Therefore p~l, A is transitive on l, and l is unique. If A -~R, assume that there is a point p fixed by A which lies on no fixed line. Then A must be transitive on p~ and as above, there is only one fixed line I of A, and A is transitive on 1. By Brouwer's theorem, A cannot be effective on po ,~ $1. Its ineffectivity subgroup Z, by definition the group of all (p, /)-dilatations in A, is therefore a nontrivial closed subgroup of A, and hence infinite cyclic. Lemma 2. If A N o , a contradiction to T h e o r e m 1. Hence I C i \ Ci[ < 1. --

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4. Tangents, Ovals, Spirals Definition. Let P be a projective plane, A _ P. A line t is called a tangent of A at p iff (1) A c~ t - - {p} (2) [A c~ g] > 2 for every other line g containing p.

Definition. Let P be a topological projective plane, A _c p. A line t is called a local tangent of A at p iff (1) p has a n e i g h b o r h o o d U with ( A n U) n t = {p} (2) t = lira p v a. a~A ct~ p

The definitions imply that tangents and local tangents, if they exist, are unique.

Definition. Let P be a projective plane, 0 _ P. 0 is called an oval iff (1) [ 0 n g l < 2 for each line g (2) 0 has a tangent at each point.

1-Dimensional Orbits in Flat Projective Planes

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Definition. Let P be a topotogicaI projective plane, S~_P. S is called a

semiovaI iff (1) ]Sc~gl