Random walks on weakly hyperbolic groups
Random walks on weakly hyperbolic groups
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DOI：
10.1515/crelle-2015-0076
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发表时间：
2014-10
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通讯作者：
Joseph Maher;G. Tiozzo
中科院分区：
文献类型：
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作者：
Joseph Maher;G. Tiozzo
设G是可数群，它通过等距作用在可分但不一定是真的Gromov双曲空间X上。我们称G的作用是弱双曲的，如果G包含两个独立的双曲等距。我们证明了在这样的G上的随机游动几乎必然收敛到Gromov边界。我们应用的收敛结果显示线性进展和线性增长的平移长度，没有任何假设的时刻的随机游动。如果作用是非圆柱的，并且随机游动具有有限的熵和有限的对数矩，我们证明了具有命中测度的Gromov边界是Poisson边界。
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.