Random walks on weakly hyperbolic groups. (English) Zbl 1434.60015
Summary: 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.
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.
MSC:
| 60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |
| 20F65 | Geometric group theory |
| 60G50 | Sums of independent random variables; random walks |
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