Gromov boundaries as Markov compacta

TL;DR

This paper demonstrates that the Gromov boundary of any hyperbolic group can be represented as a Markov compactum, providing a finite, simplicial, and bi-Lipschitz equivalent description of its natural quasi-conformal structure.

Contribution

It constructs a sequence of covers leading to a Markov compactum representation of the boundary, extending results to all finitely generated hyperbolic groups with a semi-Markovian structure.

Findings

Gromov boundary is homeomorphic to a Markov compactum.

Constructs a bi-Lipschitz equivalent metric to the visual metric.

Provides a finite simplicial description of the boundary's structure.

Abstract

We prove that the Gromov boundary of every hyperbolic group is homeomorphic to some Markov compactum. Our reasoning is based on constructing a sequence of covers of $\partial G$, which is quasi-$G$-invariant wrt. the ball $N$-type (defined by Cannon) for $N$ sufficiently large. We also ensure certain additional properties for the inverse system representing $\partial G$, leading to a finite description which defines it uniquely. By defining a natural metric on the inverse limit $\lim K_n$ and proving it to be bi-Lipschitz equivalent to an accordingly chosen visual metric on $\partial G$, we prove that our construction enables providing a simplicial description of the natural quasi-conformal structure on $\partial G$. We also point out that the initial system of covers can be modified so that all the simplexes in the resulting inverse system are of dimension less than or equal to $\dim \partial G$. We also generalize --- from the torsion-free case to all finitely generated hyperbolic groups --- a theorem guaranteeing the existence of a finite representation of $\partial G$ of another kind, namely a semi-Markovian structure (which can be understood as an analogue of the well-known automatic structure of $G$ itself).