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 $U \subset G$ containing the unipotent elements of $G$ and for all $C > 0$, the set of discrete subgroups $\Gamma \subset G$ such that (a) $\Gamma \bigcap U = \{ e\}$, (b) $G/\Gamma$ compact and measure $(G/\Gamma ) \leqq C$, is compact. As an application, for any genus $g$ and $\varepsilon > 0$, the set of compact Riemann surfaces of genus $g$ all of whose closed geodesics in the Poincaré metric have length $\geqq \varepsilon$, is itself compact.