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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A remark on Mahler’s compactness theorem


Author: David Mumford
Journal: Proc. Amer. Math. Soc. 28 (1971), 289-294
MSC: Primary 22.20
DOI: https://doi.org/10.1090/S0002-9939-1971-0276410-4
MathSciNet review: 0276410
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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.


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Keywords: Discrete subgroups, Mahler’s theorem
Article copyright: © Copyright 1971 American Mathematical Society