A Remark on Mahler's Compactness Theorem

We prove that if G is a semisimple Lie group without compact factors, then for all open sets UCG containing the unipotent elements of G and for all C>O, the set of discrete subgroups rCG such that (a) rn=U[{e}, (b) G/r compact and measure (G/r) 5 C, is compact. As an application, for any genus g and e>0, the set of compact Riemann surfaces of genus g all of whose closed geodesics in the Poincare metric have length 2 , is itself compact. Consider the following general problem: let G be a locally compact topological group and let 9a = { the set of discrete subgroups r C G[. We would like to put a good topology on WaG and we would like to find fairly "big" subsets of 9JG that turn out to be compact. Mahler studied the case G = Rn, G/r compact, i.e., r is lattice (cf. Cassels [1, Chapter 5]). In this case, the group of automorphisms of G, GL(n, R), acts transitively on the set of lattices, so that the subset WGCEG of lattices can be identified as a homogeneous space under GL(n, R); in fact: 9a _ GL (n, R) / GL (n, Z). So there is only one natural topology on Og and Mahler's theorem states that for all e and K: {rC R (1) if y E r, IIPYII < e= }=O is compact. (2) volume (Rn/r) K K (Cassels [1, p. 137 ].) Chabauty [2] has investigated generalizations of Mahler's theorem to general G and subgroups r such that measure (G/r) < + X'.1 We topologize 9JG by taking as a basis for the open sets the following: Received by the editors April 29, 1970. AMS 1970 subject classifications. Primary 22E40.