Random walks on hyperbolic groups and their Riemann surfaces

TL;DR

This paper studies random walks on hyperbolic groups, providing methods to compute probabilities, drifts, and geometric invariants like hyperbolic distance, with explicit results for the braid group B3.

Contribution

It introduces a new method using Cayley graphs to compute distributions and invariants for random elements in hyperbolic groups, including explicit calculations for B3.

Findings

Computed the probability distribution of minimal word length in hyperbolic groups

Explicitly calculated the drift and return probability for B3

Established a closed formula relating hyperbolic distance and other invariants

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.