Random walks on weakly hyperbolic groups

Abstract

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk.If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

“…A symmetric probability measure on is called nonelementary if the subgroup of G generated by its support is a nonelementary subgroup of . The following results are due in this generality to Maher and Tiozzo . Theorem Let be a countable group that acts by isometries on a separable Gromov hyperbolic space such that any two points in can be connected by a geodesic.…”

“…Using techniques of , Dahmani and Horbez [, Proposition 1.9] proved. Proposition Let be a separable geodesic Gromov hyperbolic metric space, with hyperbolicity constant .…”

“…Moreover, almost every sample path converges to some (the Gromov boundary of ), and there is a unit speed geodesic ray starting at and converging to such that . We refer to for a detailed discussion with a comprehensive list of references. Let be the coarsely well‐defined map sending to a shortest curve on .…”

“…We begin with reviewing some background on random walks on groups. This is a vast subject, see, for example, [7,15,18] for more details.…”

“…In the case that (X, d X ) is a separable Gromov hyperbolic metric space, then the action of G on X is called nonelementary if G contains a pair of loxodromic isometries with disjoint sets of fixed points in the Gromov boundary ∂X of X. A symmetric probability measure on G is called nonelementary if the subgroup of G generated by its support is a nonelementary subgroup of G. The following results are due in this generality to Maher and Tiozzo [18]. Theorem 6.1.…”