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

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.

38 pages, 1 figure. Main paper by A. Goldsborough, M. Hagen, H. Petyt and A. Sisto; appendix by J. Russell. To appear in Groups, Geometry, and Dynamics