MARKOV OPERATORS ON THE SOLVABLE BAUMSLAG–SOLITAR GROUPS

FLORIAN MARTIN AND ALAIN VALETTE

Abstract. We consider the solvable Baumslag–Solitar group

BS_{n} = ⟨a, b | aba^{−1} = b^{n}⟩,

for n ≥ 2, and try to compute the spectrum of the associated 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 both cases that SpM_{S} is connected.

For S = {a, b} (nonsymmetric case), we show that the intersection of SpM_{S} with the unit circle is the set C_{n−1} of (n−1)-st roots of 1, and that SpM_{S} contains the n − 1 circles

\{z ∈ ℂ : |z − \frac{1}{2}ω| = \frac{1}{2}\}, for ω ∈ C_{n−1},

together with the n + 1 curves given by

(\frac{1}{2}w^{k} − λ)(\frac{1}{2}w^{−k} − λ) − \frac{1}{4}exp 4πiθ = 0,

where w ∈ C_{n+1}, θ ∈ [0, 1].

Conditional on the Generalized Riemann Hypothesis (actually on Artin’s conjecture), we show that SpM_{S} also contains the circle \{z ∈ ℂ : |z| = \frac{1}{2}\}. This is confirmed by numerical computations for n = 2, 3, 5.

For S = {a^{±1}, b^{±1}} (symmetric case), we show that SpM_{S} = [−1, 1] for n odd, and SpM_{S} = [−\frac{3}{4}, 1] for n = 2. For n even, at least 4, we only get SpM_{S} = [r_{n}, 1], with

−1 < r_{n} ≤ − sin^{2} \frac{πn}{2(n + 1)}.

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