A Remark on Mahler's Compactness Theorem

Abstract

We prove that if G is a semisimple Lie group without compact factors, then for all open sets U⊂G containing the unipotent elements of G and for all C>0, the set of discrete subgroups Γ⊂G such that (a) Γ∩U={e}, (b) G/Γ compact and measure (G/Γ)≤C, is compact. As an application, for any genus g and ∈>0, the set of compact Riemann surfactes fo genus g all of whose closed geodesics in the Poincare metric have length ≥∈, is itself compact.