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Markov operators on the solvable Baumslag-Solitar groups. (English) Zbl 0983.43005

In this paper the authors consider the solvable Baumslag-Solitar group [see G. Baumslag and D. Solitar, Bull. Am. Math. Soc. 68, 199-201 (1962; Zbl 0108.02702)] \(BS_n = \langle a,b |aba^{-1} = b^n \rangle\) for \(n \geq 2\) and compute the spectrum of the associated Markov operator \(M_S\), defined on \(\ell^2(\Gamma)\) by \((M_S\xi)(x) = {1\over|S|}\sum_{s\in S}\sin(xs)\), \(\forall \xi\in \ell^2(\Gamma)\), \(\forall x\in \Gamma,\) either for the oriented Cayley graph \(\mathcal G(\Gamma,S)\) with vertex set \( \{(x,xs)\); \(x\in \Gamma\), \(s\in S\}\), and with \(S = \{a,b\}\), or for the usual Cayley graph \(\mathcal G(\Gamma,S)\), with \(S = \{a^{\pm 1}, b^{\pm 1} \}\), where \(\Gamma\) is a finitely generated group containing \(S\).
In the nonsymmetric case (Section 3) of \(S=\{a,b\}\), the authors show (Theorem 3.1) that the intersection of the spectrum \(Sp M_S\) with the unit circle is the set \(C_{n-1}\) of \((n-1)\)-th roots of 1, and that \(Sp M_S\) contains the \(n-1\) circles \(\{z\in \mathbb C\); \(|z-{1\over 2}\omega|= {1\over 2} \}\) centered at \(\omega\in C_{n-1},\) together with the \(n+1\) curves given by \(({1\over 2}w^k -\lambda)({1\over 2}w^{-k} - \lambda) - {1\over 4}\exp 4\pi i \theta = 0,\) where \(w\in C_{n+1}\), \(\theta\in [0,1]\).
In the symmetric case (Section 4) of \(S=\{a^{\pm 1}, b^{\pm 1}\}\), they show (Theorems 4.1 and 4.4) that \(Sp M_S = [-1,1]\) for odd \(n\), and \(Sp M_S = [-{3\over 4},1]\) for \(n=2\). For even \(n\geq 4\), \(Sp M_S = [r_n, 1]\), with \(-1 < r_n \leq -\sin^2{\pi n \over 2(n+1)}\).

MSC:

43A80 Analysis on other specific Lie groups
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
22E30 Analysis on real and complex Lie groups

Citations:

Zbl 0108.02702
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References:

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