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Original Articles

Markov Operators on the Solvable Baumslag–Solitar Groups

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Pages 291-300 | Published online: 04 Apr 2012
 

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 SpMs with the unit circle is the setCn–1 of (n–1)-st roots of 1, and that Sp Ms contains the n – 1 circles

together with the n + 1 curves given by

where w ∈ Cn+1, θ ∈ [0,1].

Conditional on the Generalized Riemann Hypothesis (actually on Artin's conjecture), we show that Sp Ms 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 Sp Ms = [−1,1] for n odd, and Sp Ms, = [−,¾ 1] for n = 2. For n even, at least 4, we only get Sp Ms = [rn, 1], with

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

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