Markov operators on the solvable Baumslag-solitar groups

We consider the solvable Baumslag-Solitar group BSn = < a, b / aba(-1) = b(n)>, for n greater than or equal to 2, and try to compute the spectrum of the as sociated Markov operators M-S, either for the oriented Cayley graph (S = {a , b}), or for the usual Cayley graph (S = (a(+/-1), b(+/-1)}). We show in b oth cases that Sp M-S is connected. For S = {a, b} (nonsymmetric case), we show that the intersection of Sp M-S with the unit circle is the set Cn-1 of (n-1)-st roots of 1, and that Sp M -S contains the n - 1 circles {z is an element of C : /z - 1/2w/ = 1/2}, for w is an element of Cn-1, together with the n + 1 curves given by (1/2w(k) - lambda) (1/2w(-k) - lambda) - 1/4exp4 pi i theta = 0, where w is an element of Cn+1, theta is an element of [0, 1]. Conditional on the Generalized Riemann Hypothesis (actually on Artin's conj ecture), we show that Sp M-S also contains the circle {z is an element of C : /z/ = 1/2}. This is confirmed by numerical computations for n = 2, 3, 5. For S = {a(+/-1) , b(+/-1)} (symmetric case), we show that Sp M-S = [-1,1] for n odd, and Sp M-S = [-3/4, 1] for n = 2. For n even, at least 4, we onl y get Sp M-S = [r(n), 1], with -1 < r(n) less than or equal to - sin(2) pi n/2(n+1) We also give a potential application of our computations to the theory of w avelets.