On spectra of simple random walks on one-relator groups.With an appendix by Paul Jolissain

Pierre-Alain Cherix, Alain J. Valette and Paul Jolissaint

Vol. 175 (1996), No. 2, 417–438

DOI: 10.2140/pjm.1996.175.417

Abstract

For a one relator group Γ = ⟨X : r⟩, 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 Γ respectively. We show that, generically (in the sense of Gromov), the spectral radius of h_X is (#X)^−1∕2 (which implies that the semi-group generated by X is free). We give upper bounds on the spectral radii of h_X and h_S. Finally, for Γ the fundamental group of a closed Riemann surface of genus g ≥ 2 in its standard presentation, we show that the spectrum of h_S is an interval [−r,r], with r ≤ g⁻¹(2g − 1)^1∕2. Techniques are operator-theoretic.

Mathematical Subject Classification 2000

Primary: 43A05

Secondary: 20F05, 60B15, 60J15

Milestones

Received: 15 May 1994

Revised: 5 March 1995

Published: 1 October 1996