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Proceedings of the American Mathematical Society
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
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0276410-4
PII: S 0002-9939(1971)0276410-4
Keywords: Discrete subgroups, Mahler's theorem
Article copyright: © Copyright 1971 American Mathematical Society