A remark on Mahler’s compactness theorem

Abstract

We prove that if G G is a semisimple Lie group without compact factors, then for all open sets U ⊂ G U \subset G containing the unipotent elements of G G and for all C > 0 C > 0 , the set of discrete subgroups Γ ⊂ G \Gamma \subset G such that (a) Γ ⋂ U = { e } \Gamma \bigcap U = \{ e\} , (b) G / Γ G/\Gamma compact and measure ( G / Γ ) ≦ C (G/\Gamma ) \leqq C , is compact. As an application, for any genus g g and ε > 0 \varepsilon > 0 , the set of compact Riemann surfaces of genus g g all of whose closed geodesics in the Poincaré metric have length ≧ ε \geqq \varepsilon , is itself compact.