Random Walks on Graphs and Approximation of L2-Invariants

TL;DR

This paper interprets right multiplication operators on group l2 spaces as random walks on graphs, providing a new proof and perspective on Lück's theorem regarding the approximation of l2-Betti numbers for residually finite groups.

Contribution

It introduces a novel interpretation of group operators as random walks on graphs, leading to a simplified proof of a key theorem on l2-invariants.

Findings

New proof of Lück's theorem on l2-Betti number approximation

Interpretation of group operators as random walks on graphs

Application of graph convergence and probability measure techniques

Abstract

Right multiplication operators $R_w: l_2G \rightarrow l_2G$, $w \in \C[G]$, are interpreted as random-walk operators on labelled graphs that are analogous to Cayley graphs. Applying a generalization of the graph convergence defined by R. Grigorchuk and A. \.{Z}uk \cite{Grigorchuk_Zuk_1} gives a new proof and interpretation of a special case of W. L\"{u}ck's famous Theorem on the Approximation of $l_2$-Betti numbers for countable residually finite groups. In particular, using this interpretation, the proof follows quickly from standard theorems about the weak convergence of probability measures that are characterized by their moments.