Скачать или смотреть Richard Sharp (University of Warwick)

Title: Random walks on groups, amenability and ratio limit theorems

Abstract: A famous result of Kesten from 1959 relates symmetric random walks on countable groups to amenability. Precisely, provided the support of the walk generates the group, the probability of return to the identity decays exponentially fast if and only if the group is not amenable. This led to many analogous “amenability dichotomies”, for example for the spectrum of the Laplacian of manifolds and critical exponents of discrete groups of isometries. I will discuss some of these topics and present a version of the dichotomy for non-symmetric walks. Time permitting, I will also discuss a ratio limit theorem for amenable groups. This is joint work with Rhiannon Dougall.