Geometric Structures on 3-manifolds

Let $G$ be a group acting by isometries on a Gromov hyperbolic space, which need not be proper. If $G$ contains two hyperbolic elements with disjoint fixed points, then we show that a random walk on $G$ converges to the boundary almost surely. This gives a unified approach to convergence for the mapping class groups of surfaces, $\mathrm{Out}(F_n)$ and acylindrical groups. This is joint work with Giulio Tiozzo.