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We show that quasi-isometries of (well-behaved) hierarchically hyperbolicgroups descend to quasi-isometries of their maximal hyperbolic space. This hastwo applications, one relating to quasi-isometry invariance of acylindricalhyperbolicity, and the other a linear progress result for Markov chains. Theappendix, 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