Percolation in acylindrically hyperbolic groups

TL;DR

This paper proves that in acylindrically hyperbolic groups, Bernoulli bond percolation exhibits a nonuniqueness phase with infinitely many infinite clusters, extending previous results to broader classes of groups.

Contribution

It generalizes Hutchcroft's result on Gromov hyperbolic graphs to include relatively hyperbolic groups, mapping class groups, and rank-1 CAT(0) groups.

Findings

Bernoulli bond percolation has a nonuniqueness phase in acylindrically hyperbolic groups

Existence of infinitely many infinite clusters in this phase

Extension of hyperbolic graph results to broader group classes

Abstract

Let $G$ be an acylindrically hyperbolic group. We prove that Bernoulli bond percolation on every Cayley graph of $G$ has a nonuniqueness phase, in which there are infinitely many infinite clusters. This generalizes Hutchcroft's result for Gromov hyperbolic graphs to relatively hyperbolic groups, mapping class groups and rank-1 CAT(0) groups for example.