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

Antoine Goldsborough, Mark Hagen, Harry Petyt, Jacob Russell, Alessandro Sisto

TL;DR

This work studies when quasi-isometries of groups acting on hyperbolic spaces induce quasi-isometries on a canonical hyperbolic space associated with hierarchically hyperbolic groups. It proves a descent theorem: for well-behaved HHGs, quasi-isometries of the group descend to the maximal hyperbolic space $\mathcal{C}S$, enabling a transfer of geometric properties through the HHG structure. The paper then uses projection machinery, Morse theory, and Behrstock-type inequalities to establish linear progress with exponential decay for tame, quasi-homogeneous Markov chains, with consequences for random divergence within these groups. A key application is that acylindrical hyperbolicity is, at least in this setting, quasi-isometry invariant: any group quasi-isometric to a non-elementary HHG with unbounded maximal space is acylindrically hyperbolic. An appendix by Jacob Russell provides a partial converse when $\mathcal{C}S$ is one-ended, connecting the boundary behavior to quasi-isometries of the ambient group.

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.

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

TL;DR

This work studies when quasi-isometries of groups acting on hyperbolic spaces induce quasi-isometries on a canonical hyperbolic space associated with hierarchically hyperbolic groups. It proves a descent theorem: for well-behaved HHGs, quasi-isometries of the group descend to the maximal hyperbolic space , enabling a transfer of geometric properties through the HHG structure. The paper then uses projection machinery, Morse theory, and Behrstock-type inequalities to establish linear progress with exponential decay for tame, quasi-homogeneous Markov chains, with consequences for random divergence within these groups. A key application is that acylindrical hyperbolicity is, at least in this setting, quasi-isometry invariant: any group quasi-isometric to a non-elementary HHG with unbounded maximal space is acylindrically hyperbolic. An appendix by Jacob Russell provides a partial converse when is one-ended, connecting the boundary behavior to quasi-isometries of the ambient group.

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.
Paper Structure (21 sections, 43 theorems, 70 equations, 2 figures)

This paper contains 21 sections, 43 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and assumptions
  3. Geometric group theory notions and definitions
  4. Assumptions on the group action
  5. Projections
  6. Markov chains
  7. Incompatible Morse rays
  8. Bounded projections to axes
  9. Linear progress
  10. Notation and preliminary lemmas
  11. Proof of Proposition \ref{['prop:axiom']}
  12. Descent of quasi-isometries in HHSs
  13. Acylindrical hyperbolicity and quasi-isometries
  14. Application to hierarchically hyperbolic groups
  15. Appendix: Quasi-isometries of HHGs with one-ended maximal hyperbolic spaces, by Jacob Russell
  16. ...and 6 more sections

Key Result

Theorem 1

Let $(G,\mathfrak{S})$ be a well-behaved HHG, with maximal hyperbolic space $\mathcal{C} S$, and let $\pi_S:G\to \mathcal{C} S$ be the associated projection. Every quasi-isometry $f:G\to G$ induces a quasi-isometry $\bar{f}:\mathcal{C} S\to \mathcal{C} S$ such that $\pi_S f$ and $\bar{f}\pi_S$ coars

Figures (2)

  • Figure 1: For a contradiction, we assume that with overwhelming probability the Markov chain creates a large projection on $hE(g)$. The dotted line is meant to represent the sample path of the Markov chain and $\alpha$, $\beta$ are the two Morse rays with very different Morse gauges.
  • Figure 2: If the leftmost graph is part of $\Gamma_\mathfrak{S}$ (with non-adjacent vertices transverse), then it yields the second graph in $\Gamma_{\mathfrak{S}\smallsetminus\mathcal{V}}$, which is obtained by applying Proposition \ref{['prop:qis_descend_once:biinfinite']}. Repeating yields the third graph, and we terminate with the fourth graph, which has vertices representing non-maximal unbounded domains. Hence $\widehat{X}$ is not quasiisometric to $\mathcal{C} S$ in this case.

Theorems & Definitions (91)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.7: Tame
  • Definition 2.8: Quasi-homogeneous
  • Remark 2.9
  • ...and 81 more