  We propose the study of Markov chains on groups as a "quasi-isometry
invariant" theory that encompasses random walks. In particular, we focus on
certain classes of groups acting on hyperbolic spaces including
(non-elementary) hyperbolic and relatively hyperbolic groups, acylindrically
hyperbolic 3-manifold groups, as well as fundamental groups of certain graphs
of groups with edge groups of subexponential growth. For those, we prove a
linear progress result and various applications, and these lead to a Central
Limit Theorem for random walks on groups quasi-isometric to the ones we
consider.
