The Spectra of Lamplighter Groups and Cayley Machines

2004/12/20 by Mark Kambites, Kambites, Mark, Pedro V. Silva +3
Computer Science · Mathematics · #20E08 #20E22 #20F38 #Cellular Automata and Applications #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Spectral Theory (math.SP) #math.GR #math.SP #msc:20E08 #msc:20E22 #msc:20F38

paper · pdf · doi:10.48550/arxiv.math/0412412

36 pages, improved exposition, main results slightly strengthened

openalex publication_date 2004/12/20 · arxiv created 2005/09/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We calculate the spectra and spectral measures associated to random walks on restricted wreath products of finite groups with the infinite cyclic group, by calculating the Kesten-von Neumann-Serre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk and Zuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by automata coincides with the Kesten-von Neumann-Serre spectral measure.

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