# Growth of balls in the universal cover of graphs & surfaces

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Steve Karam

created by risa on 20 Feb 2015

25 feb 2015
-- 14:30

Aula di Consiglio, Dip. Matematica, Università "La Sapienza", Roma

**Abstract.**

We prove that if the area of a closed Riemannian surface M of genus at least two is sufficiently small with respect to its hyperbolic area, then for every radius R≥0 the universal cover of M contains an R-ball with area at least the area of a cR-ball in the hyperbolic plane, where c is a universal positive constant in (0,1). In particular (taking the area of M smaller if needed), we prove that for every radius R ≥ 1, the universal cover of M contains an R-ball with area at least the area of a ball with the same radius in the hyperbolic plane. This result answers positively a question of L. Guth for surfaces. We also prove an analog result for graphs. Specifically, we prove that if G is a connected metric graph of first Betti number b ≥ 2 and of length suffciently small with respect to the length of a connected trivalent graph Gof the same Betti number where the length of each edge is 1, then for every radius R ≥ 0 the universal cover of Gb contains an R-ball with length at least c times the length of an R-ball in the universal cover of Gb where c is a universal constant in (0.5,1).