On the spectrum of Hecke type operators related to some fractal groups (Q2783047)

This is a half-research, half-survey paper. The most striking original result is the first construction of a 4-regular graph \(X\) such that the spectrum of the adjacency matrix of \(X\) is a Cantor set. This graph \(X\) is obtained as follows: let a certain group \(G\) of intermediate growth (previously introduced by the second author [see\textit{R. I. Grigorchuk} [Funct. Anal. Appl. 14, 41--43 (1980); translation from Funkts. Anal. Prilozh. 14, No. 1, 53--54 (1980; Zbl 0595.20029)]) act on the rooted binary tree \(T_{2}\). Let \(P\) be the stabilizer in \(G\) of some infinite ray in \(T_{2}\). Then \(X\) is the Schreier graph associated with the homogeneous space \(G/P\). The Cantor spectrum of \(X\) is obtained as \(F(J)\), where \(F\) is a simple algebraic function and \(J\) is the Julia set of the quadratic map \(z\mapsto z^{2}-\lambda\), for a specific \(\lambda\).NEWLINENEWLINEBesides these (and other related) original results, the paper also provides excursions through unitary representations, dynamical systems, automata and Hecke operators. The paper concludes with a list of open questions.NEWLINENEWLINEReviewer's remark: The final open question is somewhat heretical in view of the present paradigm: It hints at a finitely generated torsion-free subgroup of \(\mathrm{Aut}(T)\), with \(T\) a rooted tree, such that the adjacency matrix of the Cayley graph has a gap in its spectrum, or even has totally disconnected spectrum: such a group would disprove the celebrated Baum-Connes conjecture [see \textit{P. Baum, A. Connes} and \textit{N. Higson}, Contemp. Math. 167, 241--291 (1994; Zbl 0830.46061)].NEWLINENEWLINEFor the entire collection see [Zbl 0981.00006].