2016/01/14 by Joseph Maher, Giulio Tiozzo · 1 citation
paper · doi:10.1515/crelle-2015-0076
crossref issued 2016/01/14 · crossref published 2016/01/14 · crossref published-online 2016/01/14 · crossref created 2018/08/31 · crossref published-print 2018/09/01 · crossref deposited 2025/07/06 · crossref indexed 2026/08/03
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.