Random walks on Convergence Groups

Aitor Azemar

Random walks on Convergence Groups

Geometric Topology 2020-06-16 v3

Authors: Aitor Azemar

Abstract

We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular we prove that if a convergence group $G$ acts on a compact metrizable space $M$ with the convergence property then we can provide $G\cup M$ with a compact topology such that random walks on $G$ converge almost surely to points in $M$ . Furthermore we prove that if $G$ is finitely generated and the random walk has finite entropy and finite logarithmic moment with respect to the word metric, then $M$ , with the corresponding hitting measure, can be seen as a model for the Poisson boundary of $G$ .

Keywords

random walk in random environment random walk on graphs topological groups

Cite

@article{arxiv.1810.09486,
  title  = {Random walks on Convergence Groups},
  author = {Aitor Azemar},
  journal= {arXiv preprint arXiv:1810.09486},
  year   = {2020}
}

Comments

Extended results to more general measures and general cleanup. Theorem enumeration has changed between v2 and v3. A corollary of the preprint had already been proven by Gekhtman, Gerasimov, Potyagailo and Yang. Added a reference to the paper where it is proven (arXiv:1708.02133)

Random walks on Convergence Groups

Abstract

Keywords

Cite

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