Characterization of hyperbolic groups via random walks

TL;DR

This paper characterizes hyperbolic groups using properties of random walks, providing new probabilistic criteria that distinguish hyperbolic groups from others, and extends classical results like Ancona's theorem.

Contribution

It introduces novel probabilistic conditions involving random walks that characterize hyperbolic groups, including a partial converse to Ancona's theorem and an if-and-only-if criterion.

Findings

Hyperbolic groups satisfy specific random walk intersection probabilities.

Nonamenable groups with certain random walk properties are hyperbolic.

A geometric criterion for hyperbolicity of graphs is established.

Abstract

Our first result gives a partial converse to a well-known theorem of A. Ancona for hyperbolic groups. We prove that a group $G$, equipped with a symmetric probability measure whose finite support generates $G$, is hyperbolic if it is nonamenable and satisfies the following condition: for a sufficiently small $\varepsilon >0$ and $r\geqslant0$, and for every triple $(x, y, z)$, belonging to a word geodesic of the Cayley graph, the probability that a random path from $x$ to $z$ intersects the closed ball of radius $r$ centered at $y$ is at least $1-\varepsilon.$ We note that if a group is hyperbolic then the above condition for $r=0$ is satisfied by Ancona's theorem and for any $r>0$ follows from this paper. Another our theorem claims that a finitely generated group is hyperbolic if and only if the probability that a random path, connecting two antipodal points of an open ball of radius $r$ does not intersect it is exponentially small with respect to $r$ for $r\gg0$.. The proof is based on a purely geometric criterion for the hyperbolicity of a connected graph.