Sublinear deviation between geodesics and sample paths

Abstract

We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmüller distance for random walks on mapping class groups, and on Cayley graphs of a large class of finitely generated groups.

Random walks on weakly hyperbolic groups

“…We then show the sublinear tracking results, using work of Tiozzo [Tio12]. Sublinear tracking can be thought of as a generalization of Oseledec's multiplicative ergodic theorem [Ose68].…”

“…Geodesic tracking. We will now prove Theorem 1.3, using the following sublinearity result from Tiozzo [Tio12].…”

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

“…The analog of Theorem for the action of the mapping class group on is due to Tiozzo [, Theorem 18]. Theorem Let be a nonelementary probability measure on with finite first moment for .…”

“…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 .…”

“…Let V ⊂ PML be an open set. By minimality, g∈G gV = PML, and hence ν( g∈G gV ) = 1 > 0 whence ν(gV ) > 0 for some g ∈ G. By stationarity, ν(V ) > 0.The analog of Theorem 6.1 for the action of the mapping class group on T (S) is due to Tiozzo[30, Theorem 18]. Let μ be a nonelementary probability measure on Mod(S) with finite first moment for d T .…”