Markov operators on the solvable Baumslag-Solitar groups
Martin, Florian ; Valette, Alain
Experiment. Math., Tome 9 (2000) no. 3, p. 291-300 / Harvested from Project Euclid
We consider the solvable Baumslag--Solitar group ¶ $$\BS_{n}=\$$, ¶ for $n\geq 2$, and try to compute the spectrum of the associated Markov operators $M_{S}$, either for the oriented Cayley graph ($S=\{a,b\}$), or for the usual Cayley graph ($S=\{a^{\pm1},b^{\pm1}\}$). We show in both cases that $\Sp M_{S}$ is connected. ¶ For S={a,b} (nonsymmetric case), we show that the intersection of $\Sp M_{S}$ with the unit circle is the set $C_{n-1}$ of $(n{-}1)$-st roots of 1, and that $\Sp M_{S}$ contains the $n-1$ circles ¶ $$\{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
Publié le : 2000-05-14
Classification:  37A30,  05C25,  11R42,  37A45,  60G50
@article{1045952352,
     author = {Martin, Florian and Valette, Alain},
     title = {Markov operators on the solvable Baumslag-Solitar groups},
     journal = {Experiment. Math.},
     volume = {9},
     number = {3},
     year = {2000},
     pages = { 291-300},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1045952352}
}
Martin, Florian; Valette, Alain. Markov operators on the solvable Baumslag-Solitar groups. Experiment. Math., Tome 9 (2000) no. 3, pp.  291-300. http://gdmltest.u-ga.fr/item/1045952352/