Growth of balls in the universal cover of surfaces and graphs

TL;DR

This paper establishes uniform lower bounds on volume growth of balls in universal covers of surfaces and graphs, confirming a conjecture for surfaces and extending to graphs.

Contribution

It provides the first uniform lower bounds on volume growth in universal covers of surfaces and graphs, answering Guth's question for surfaces.

Findings

Universal covers of certain surfaces contain large balls with hyperbolic volume growth.

The results apply to surfaces with metrics close to hyperbolic in area.

Analogous volume growth bounds are proved for graphs.

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 $\area(M,h)\leq \delta \area(M,hyp)$ 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.