A numerical lower bound for the spectral radius of random walks on surface groups

Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus $2$ surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi.