2025/07/29 by Gerasimov, Victor, Potyagailo, Leonid
#05C81 #20F65 #20F67 #60B15 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2507.22005
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 ε >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-ε. 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≫0.. The proof is based on a purely geometric criterion for the hyperbolicity of a connected graph.