On spectra of simple random walks on one-relator groups

For a one relator group Gamma = [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 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 semi-group 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 greater than or equal to 2 in its standard presentation, we show that the spectrum of h(S) is an interval [-r, r], with r less than or equal to g(-1)(2g - 1)(1/2). Techniques are operator-theoretic.