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Growth of balls in the universal cover of surfaces and graphs


Author: Steve Karam
Journal: Trans. Amer. Math. Soc. 367 (2015), 5355-5373
MSC (2010): Primary 53C23
DOI: https://doi.org/10.1090/S0002-9947-2015-06189-3
Published electronically: April 1, 2015
MathSciNet review: 3347175
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Abstract: In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant $ \delta >0$ such that if $ (M,hyp)$ is a closed hyperbolic surface and $ h$ another metric on $ M$ with $ \mathrm {Area}(M,h)\leq \delta \mathrm {Area}(M,hyp)$, then for every radius $ R\geq 1$ the universal cover of $ (M,h)$ contains an $ R$-ball with area at least the area of an $ R$-ball in the hyperbolic plane. This positively answers a question of L. Guth for surfaces. We also prove an analog theorem for graphs.


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Steve Karam
Affiliation: Laboratoire de Mathématiques et de Physique Théorique, UFR Sciences et Technologie, Université François Rabelais, Parc de Grandmont, 37200 Tours, France
Email: steve.karam@lmpt.univ-tours.fr

DOI: https://doi.org/10.1090/S0002-9947-2015-06189-3
Received by editor(s): April 17, 2013
Published electronically: April 1, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.