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. 1. Main algorithm Let Γ be a countable group, generated by a finite symmetric set S of cardinality |S|. The simple random walk X0,X1, . . . on Γ is defined by X0 = e the identity of Γ, and Xn+1 = Xns with probability 1/ |S| for any s ∈ S. A crucial numerical parameter of this random walk is its spectral radius ρ = limP(X2n = e) 1/2n. Equivalently, denote by Wn the number of words of length n in the generators that represent e in Γ, then P(Xn = e) = Wn/ |S|, so that ρ = limW 1/2n 2n / |S|. It is equivalent to study the spectral radius or the cogrowth limW 1/2n 2n . The spectral radius is at most 1, and ρ = 1 if and only if Γ is amenable. In the free group with d generators, the generating function ∑ Wnz n can be computed explicitly (it is algebraic), and the exact value of the spectral radius follows: ρ = √ 2d− 1/d. Since words that reduce to the identity in the free group also reduce to the identity in any group with the same number of generators, one infers that in any group Γ, ρ > 2 √ |S| − 1/ |S|. Moreover equality holds if and only if the Cayley graph of Γ is a tree [Kes59]. In general, there are no explicit formulas for ρ, and even giving precise numerical estimates is a delicate question. In this short note, we will describe an algorithm giving bounds from below on ρ in some classes of groups, particularly for the fundamental group Γg of a compact surface of genus g > 2, given by its usual presentation (1.1) Γg = 〈a1, . . . , ag, b1, . . . , bg | [a1, b1] · · · [ag, bg] = e〉. Since there are 4 generators in Γ2, the above trivial bound obtained by comparison to the free group gives ρ > 0.661437. Our main estimate is the following result. Theorem 1.1. In the surface group Γ2, one has ρ > 0.662772. This improves on the previously best known result, due to Bartholdi [Bar04], giving ρ > ρBar = 0.662421. 1 Bartholdi’s method is to study a specific class of paths from the Date: June 6, 2014.