Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity (with an appendix by Jacob Russell)
Antoine Goldsborough
Heriot-Watt University, Edinburgh, UKMark F. Hagen
University of Bristol, Bristol, UKHarry Petyt
University of Oxford, Oxford, UKAlessandro Sisto
Heriot-Watt University, Edinburgh, UK

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.
Cite this article
Antoine Goldsborough, Mark F. Hagen, Harry Petyt, Alessandro Sisto, Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity (with an appendix by Jacob Russell). Groups Geom. Dyn. (2025), published online first
DOI 10.4171/GGD/873