Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity (with an appendix by Jacob Russell)

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.