Markov Operators on the Solvable Baumslag–Solitar Groups

Abstract

We consider the solvable Baurnslag-Solitar group

for n ≥ 2, and try to compute the spectrum of the associated Markov operators Ms, either for the oriented Cayley graph (5 = {a, bj), or for the usual Cayley graph (S = {a^±l, b^+l}). We show in both casesthat Sp Ms is connected.

For S = {a, b} (nonsymmetric case), we show that the intersection of S_pM_s with the unit circle is the setC_n–1 of (n–1)-st roots of 1, and that S_p M_s contains the n – 1 circles

together with the n + 1 curves given by

where w ∈ C_n+1, θ ∈ [0,1].

Conditional on the Generalized Riemann Hypothesis (actually on Artin's conjecture), we show that S_p M_s also contains the circle {z ∈ C : |z| = ½}. This is confirmed by numerical computations for n = 2,3,5.

For S = {a^±l, b^±l} (symmetric case), we show that S_p M_s = [−1,1] for n odd, and S_p M_s, = [−,¾ 1] for n = 2. For n even, at least 4, we only get S_p M_s = [r_n, 1], with

We also give a potential application of our computations to the theory of wavelets.