On spectra of simple random walks on one-relator groups. With an appendix by Paul Jolissaint
type: Item
instance of: scholarly article
title: On spectra of simple random walks on one-relator groups. With an appendix by Paul Jolissaint
zbMATH Open document ID: 0865.60059
author: Pierre-Alain Cherix
author: Alain Valette
published in: Pacific Journal of Mathematics
publication date: 1996-12-15T00:00:00Z
review text: For a one-relator group \(\Gamma= \langle X:r \rangle\), we study the spectra of the transition operators \(h_X\) and \(h_S\) associated with the simple random walks on the directed Cayley graph and ordinary Cayley graph of \(\Gamma\), respectively. We show that, generically (in the sense of Gromov), the spectral radius of \(h_X\) is \(|X|^{- 1/2}\) (which implies that the semigroup generated by \(X\) is free). We give upper bounds on the spectral radii of \(h_X\) and \(h_S\). Finally, for \(\Gamma\) the fundamental group of a closed Riemann surface of genus \(g\geq 2\) in its standard presentation, we show that the spectrum of \(h_S\) is an interval \([-r, r]\), with \(r\leq g^{-1} (2g- 1)^{1/2}\). Techniques are operator-theoretic.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 950181
zbMATH Keywords: one-relator group
zbMATH Keywords: spectra of the transition operators
zbMATH Keywords: simple random walks
zbMATH Keywords: directed Cayley graph
zbMATH Keywords: ordinary Cayley graph
zbMATH Keywords: spectral radius
zbMATH Keywords: fundamental group
zbMATH Keywords: closed Riemann surface
MaRDI profile type: Publication
full work available at URL: pjm.1996.175.417
OpenAlex ID: W1546723408
DOI: 10.2140/PJM.1996.175.417
description: scientific article; zbMATH DE number 950181