Random walks on graphs and approximation of L² - invariants

Abstract

In this work, we interpret right multiplication operators Rw : L²(G) --> L²(G), w Є C[G] as random walk operators on certain labelled graphs we employ that are analogous to Cayley graphs. Applying a generalization of the graph convergence defined by R. I. Grigorchuk and A. Zuk to these graphs gives a new interpretation and proof of a special case of W. Luck's famous Theorem on the Approximation of L²-Betti numbers for countable residually finite groups by means of exhausting towers of finite-index subgroups. In particular, using this interpretation, the theorem follows naturally from standard theorems in probability theory concerning the weak convergence of probability measures that are characterized by their moments. This paper is mainly a direct adaptation of the ideas of Grigorchuk, Zuk and Luck to this setting. We aim to explain how these ideas are related and give a short exposition of them.