Growth of balls in the universal cover of surfaces and graphs

Karam, Steve

doi:10.1090/S0002-9947-2015-06189-3

Growth of balls in the universal cover of surfaces and graphs
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Trans. Amer. Math. Soc. 367 (2015), 5355-5373 Request permission

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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