Markov operators on the solvable Baumslag-Solitar groups

Abstract

We consider the solvable Baumslag--Solitar group

$$\BS_{n}=\<a,b\mid aba^{-1}=b^{n}\>$$,

for $n\geq 2$, 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 ($S=\{a^{\pm1},b^{\pm1}\}$). We show in both 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 $C_{n-1}$ of $(n{-}1)$-st roots of 1, and that $\Sp M_{S}$ contains the $n-1$ circles

$$\{z\in\bbC:|z-\half{\omega}|=\half\},\quad\hbox{for $\omega\in C_{n-1}$}$$,

together with the $n+1$ curves given by

$$\bigl(\half{w^k}-\lambda\bigr)\bigl(\half{w^{-k}}-\lambda\bigr)-\tfrac14{\exp{4\pi i\theta}}=0,$$ where $w\in C_{n+1}$, $\theta\in [ 0,1]$.

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

For $S=\{a^{\pm1},b^{\pm1}\}$ (symmetric case), we show that $\Sp M_{S}=[-1,1]$ for n odd, and $\Sp M_{S}=[-\frac{3}{4},1]$ for n=2. For n even, at least 4, we only get $\Sp M_{S}=[r_{n},1]$, with

$$-1<r_{n}\leq -\sin^{2}\frac{\pi n}{2(n+1)}\hbox{.}$$

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