On the Lattice of Boundaries and the Entropy Spectrum of Hyperbolic Groups

TL;DR

This paper investigates the structure and spectrum of boundary actions and Furstenberg entropies in hyperbolic groups, revealing infinitely many boundaries with distinct entropies and analyzing entropy properties for free groups.

Contribution

It establishes the existence of infinitely many boundaries with distinct entropies in hyperbolic groups and analyzes entropy spectra for free groups, advancing understanding of boundary structures.

Findings

Hyperbolic groups have infinitely many boundaries with distinct entropies.

For free groups, many boundaries have entropy close to the Poisson boundary.

The entropy spectrum is shown to be closed under mild conditions.

Abstract

Let $\Gamma$ be a non-elementary hyperbolic group and $\mu$ be a probability on $\Gamma$. We study the $\mu$-proximal, stationary actions, also known as boundary actions, of $\Gamma$. In particular, we are interested in the spectrum of Furstenberg entropies of $(\Gamma,\mu)$-boundaries, and the lattice-theoretic and topological structure of the set $\mathcal{BL}(\Gamma,\mu)$ of boundaries. We prove that all hyperbolic groups have infinitely many distinct boundaries, which attain an infinite set of distinct entropies. Additionally, for simple random walks on non-abelian free groups $F_d$, we establish that there are infinitely many boundaries whose entropy is greater than $\frac{1}{2}-\epsilon$ times the entropy of Poisson boundary, when the rank $d$ is large. General results of independent interest about the order-theoretic and continuity properties of Furstenberg entropy for countable groups are attained along the way. This includes the result that under mild assumptions, the spectrum of boundary entropies $\mathcal{H}_{\text{bound}}(\Gamma,\mu)$ is closed.