Renewal theory for random walks on surface groups

We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enters a particular type of a cone and never leaves it again. As a consequence, the trajectory of the random walk can be expressed as an “aligned union” of i.i.d. trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.