Random walks on weakly hyperbolic groups

2014/10/15 by Joseph Maher, Maher, Joseph, Giulio Tiozzo +1 · 16 citations
Mathematics · #20F67 #57M60 #60G50 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #math.DS #math.GT #msc:20F67 #msc:57M60 #msc:60G50

paper · pdf · doi:10.48550/arxiv.1410.4173

58 pages, 9 figures

arxiv created 2015/01/02 · arxiv updated 2015/01/05

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.

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• Random walks on weakly hyperbolic groups
• Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces
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• Random walks on Convergence Groups
• Central Limit Theorems for Gromov Hyperbolic Groups