Random Walks on Graphs and Approximation of L2-Invariants

2016/07/27 by Zenas Wong, Wong, Zenas, Andrew Kricker +1
Computer Science · Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1607.08013

openalex publication_date 2016/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Right multiplication operators R_w: l₂G → l₂G, w ∈ \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. Żuk \citeGrigorchuk_Zuk₁ gives a new proof and interpretation of a special case of W. Lück's famous Theorem on the Approximation of l₂-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.

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