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

Abstract

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.

v2: added a geometric interpretation of the lower bound

References in corpus (1)

• On the cogrowth of Thompson's group $F$

Cited by in corpus (3)

• Density of states of tight-binding models in the hyperbolic plane
• Spectral and Combinatorial Aspects of Cayley-Crystals
• An invitation to trees of finite cone type: random and deterministic operators