Abstract: A considerable amount of research has been done on random walks on various groups. A recent development (Maher-Tiozzo 2015) has been to establish linear progress of a random walk on a weakly hyperbolic group, in the hyperbolic space X on which the group acts. In this talk, we will look at the more general case of a Markov chain (where equidistribution is no longer assumed) in specific groups acting on a hyperbolic space. In this new setting, it is possible to establish a similar linear progress result. This allows one to consider a random walk on a group H where we know that H is quasi-isometric to one of our specific groups G and establishing a Central Limit Theorem for this random walk. We also briefly mention other results that follow from linear progress of the Markov chain. This work has been carried out under the supervision of Alessandro Sisto.