Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity

TL;DR

This paper demonstrates how quasi-isometries of hierarchically hyperbolic groups induce quasi-isometries of their hyperbolic spaces, with applications to acylindrical hyperbolicity invariance and Markov chain progress.

Contribution

It establishes the descent of quasi-isometries to hyperbolic spaces and explores implications for acylindrical hyperbolicity and Markov chains.

Findings

Quasi-isometries of hierarchically hyperbolic groups descend to their hyperbolic spaces.

Quasi-isometry invariance of acylindrical hyperbolicity is established.

Linear progress for Markov chains in hyperbolic spaces is demonstrated.

Abstract

We show that quasi-isometries of (well-behaved) hierarchically hyperbolic groups descend to quasi-isometries of their maximal hyperbolic space. This has two applications, one relating to quasi-isometry invariance of acylindrical hyperbolicity, and the other a linear progress result for Markov chains. The appendix, by Jacob Russell, contains a partial converse under the (necessary) condition that the maximal hyperbolic space is one-ended.