ON SPECTRA OF SIMPLE RANDOM-WALKS ON ONE-RELATOR GROUPS

For a one relator group Gamma = [X : r], we study the spectra of the t ransition operators h(X) and h(S) associated with the simple random wa lks on the directed Cayley graph and ordinary Cayley graph of Gamma re spectively. We show that, generically (in the sense of Gromov), the sp ectral 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 o f 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 oper ator-theoretic.