Branching random walks on relatively hyperbolic groups

Consider the branching random walk on a non-elementary relatively hyperbolic group $\Gamma$ with mean offspring $r$ and base motion given by the random walk with a finitely supported and symmetric step distribution.  It is known that for $r$ in a certain interval, the population survives forever, but eventually vacates every finite subset of $\Gamma$.  We prove that in this regime, the growth rate of the BRW is equal to the growth rate $\omega_\Gamma(r)$ of the Green function of the underlying random walk.  We also prove that the Hausdorff dimension of the limit set, which is the random subset of the Bowditch boundary consisting of all accumulation points of the BRW, is equal to a constant times $\omega_\Gamma(r)$.  The talk is based on a joint work with Matthieu Dussaule and Wenyuan Yang.