Structure, expansion and probability in vertex-transitive graphs

MATX1101

Celebrated theorems of Gromov, Trofimov and Coulhon-Saloff-Coste combine to give a remarkable dichotomy for vertex-transitive graphs: such graphs must either resemble highly structured Cayley graphs, or must exhibit "expansion" in a certain sense. This in turn has had a number of striking applications, particularly to probability, such as Varopoulos's famous characterisation of those transitive graphs on which the random walk is recurrent (i.e. eventually returns to its starting point with probability 1). I will describe a number recent quantitative, finitary refinements of these results that allow us to give meaningful extensions of results like Varopoulos's to finite transitive graphs and finite regions of infinite transitive graphs.