On spectra of simple random walks on one-relator groups. With an appendix by Paul Jolissaint
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.
scientific article; zbMATH DE number 950181