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Random walks on hyperbolic groups and their Riemann surfaces

Sergei Nechaev·Raphael Voituriez
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Abstract

We investigate invariants for random elements of different hyperbolic groups. We provide a method, using Cayley graphs of groups, to compute the probability distribution of the minimal length of a random word, and explicitly compute the drift in different cases, including the braid group $B_3$. We also compute in this case the return probability. The action of these groups on the hyperbolic plane is investigated, and the distribution of a geometric invariant, the hyperbolic distance, is given. These two invariants are shown to be related by a closed formula.

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Sergei Nechaev, Raphael Voituriez. 2000-12-20. Random walks on hyperbolic groups and their Riemann surfaces. https://arxiv.org/abs/math-ph/0012037

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