Characterization of hyperbolic groups via random walks

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.