Cantor spectra and torsion-free multi-band spectra of Cayley graphs: two open existence questions for adjacency operators of groups

For a finitely generated group $\Gamma$ with a finite symmetric generating set $S$, the adjacency operator of the Cayley graph ${\rm Cay}(\Gamma,S)$ acts on $\ell^2(\Gamma)$ and its spectrum is a compact subset of the real line. Pierre de la Harpe (2024) proved that pure band spectra with any finite number $N\ge 1$ of disjoint intervals occur for Cayley graphs, but all his examples use groups with torsion. His paper reiterates, from earlier work of Dudko and Grigorchuk (2020) on the possible shapes of the spectrum of a group, the open question (Q1) whether there exists a Cayley graph whose adjacency spectrum is a Cantor set, and it leaves open (Q2) whether there exists a torsion-free group with a Cayley graph whose adjacency spectrum is a disjoint union of two or more intervals. A positive answer to Q2 would produce a non-trivial projection in the reduced group C*-algebra of a torsion-free group, refuting the Kadison–Kaplansky conjecture for that group. No construction answering either question and no impossibility theorem is known through mid-2026, so the problem is open.

Background

Let $\Gamma$ be a countable group and $S$ a finite symmetric generating set (that is, $S=S^{-1}$), so that the Cayley graph ${\rm Cay}(\Gamma,S)$ has vertex set $\Gamma$ and an edge joining $g$ and $gs$ for each $g\in\Gamma$, $s\in S$; when $S$ does not contain the identity this is a simple graph of bounded degree $|S|$. The adjacency operator $A$ is the bounded self-adjoint operator on the Hilbert space $\ell^2(\Gamma)$ given by $(A\xi)(g)=\sum_{s\in S}\xi(gs)$, and its spectrum $\Sigma(A)$ is a non-empty compact subset of $\mathbb{R}$. For Cayley graphs the discrete Laplacian and the Markov operator (the transition operator of the simple random walk, normalized so that all generators are equally likely) are affine images of $A$, so the shape of the spectrum as a set is the same question for all three operators. Following de la Harpe, one says that $A$ has a pure band spectrum with $N$ bands if $\Sigma(A)$ is a disjoint union of exactly $N$ closed intervals $\bigcup_{j=1}^{N}[a_j,b_j]$ with $N\ge 1$ and $a_1<b_1<a_2<\cdots<a_N<b_N$ (a single interval is the case $N=1$); a Cantor set spectrum means that $\Sigma(A)$ is a non-empty, perfect, nowhere dense compact subset of $\mathbb{R}$.

The study of such spectra goes back to Kesten (1959), who showed that the spectrum is a single interval for free abelian groups and for finitely generated free groups (the free group on $m$ generators with its standard generators gives the interval $[-2\sqrt{2m-1},\,2\sqrt{2m-1}]$ for the adjacency operator). Grigorchuk and Żuk (2001) computed the spectrum of the lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ (the wreath product, i.e. the semidirect product of a countable direct sum of copies of the two-element group, the 'lamps', with the integers, the 'cursor') for the generating set arising from a two-state automaton: the spectrum is again the single interval $[-1,1]$ for the Markov operator, with a spectral measure that is discrete, concentrated on a countable dense set of points; Lehner, Neuhauser and Woess (2008) generalized this to arbitrary switch-walk-switch walks on wreath products via percolation. Two-band spectra first appeared for the Grigorchuk groups $G_\omega$ generated by four involutions, whose Markov spectrum is $[-1/2,0]\cup[1/2,1]$ (Dudko–Grigorchuk 2020), and for uncountable families of groups acting on the binary rooted tree (Grigorchuk–Nagnibeda–Pérez 2022). Kuhn (1992) had earlier produced, for every $n\ge 3$, Markov operators with an $n$-band spectrum on a group, but with non-uniform transition probabilities, which therefore do not come from the adjacency operator of a Cayley graph.

In this landscape, Dudko and Grigorchuk (2020) asked what shapes the spectrum of a group can have at all, noting that it was unknown whether it could be a union of $n\ge 3$ disjoint intervals, a countable set of points accumulating to a point, or a Cantor set. Two parts of this program have since been settled: Grigorchuk and Simanek (2021) found a generating set of the lamplighter group for which the Cayley-graph spectrum is an interval together with a countable set of isolated points accumulating to a point outside it (the first example with infinitely many spectral gaps), and de la Harpe (2024), using sums and products of graphs and the classical description of spectra of tensor sums of self-adjoint operators, proved that for every $N\ge 1$ there are groups with Cayley graphs whose adjacency spectrum is a disjoint union of $N$ intervals, realizing band spectra with arbitrarily many bands for the first time. His examples are Cayley graphs of products $\mathbb{Z}^d\times\Gamma_1\times\cdots\times\Gamma_N$ with finite factors $\Gamma_i$, hence groups with torsion (elements of finite order). Cantor-type spectra are known only outside the class of Cayley graphs: Grigorchuk, Nagnibeda and Pérez (2022) exhibited infinite Schreier graphs (graphs of the action of a group on cosets of a subgroup, as opposed to the regular action on itself that defines a Cayley graph) whose spectrum is a Cantor set of Lebesgue measure zero together with countably many isolated points.

The torsion-free side is obstructed by operator algebras. For a countable group $\Gamma$, the reduced group C*-algebra $C^*_r(\Gamma)$ is the norm-closure of the algebra of convolution operators on $\ell^2(\Gamma)$; the adjacency operator of any Cayley graph of $\Gamma$ lies in $C^*_r(\Gamma)$. If a spectrum splits into two or more disjoint closed pieces, then by the holomorphic functional calculus the spectral projection onto one piece is a non-trivial projection (a self-adjoint idempotent) in $C^*_r(\Gamma)$. The Kadison–Kaplansky conjecture asserts that $C^*_r(\Gamma)$ has no non-trivial projections when $\Gamma$ is torsion-free; it is known for large classes of groups, including torsion-free amenable groups (via the Baum–Connes conjecture, Higson–Kasparov 2001), word-hyperbolic groups (Puschnigg 2002), and subgroups of hyperbolic groups (Mineyev–Yu 2002), and remains open in general, with no counterexample known. Hence a torsion-free group with a multi-band Cayley spectrum would be a counterexample to this conjecture, while Dudko–Grigorchuk (2020) did construct torsion-free (even step-3 solvable) groups with a one-band spectrum $[-1,1]$.

Problem Statement

Decide the two existence questions left open by de la Harpe's 2024 construction of Cayley graphs with pure band spectra, which together form the surviving core of the Dudko–Grigorchuk problem on the possible shapes of the spectrum of a group (equivalently formulated for the discrete Laplacian or the Markov operator of the simple random walk, since these are affine images of the adjacency operator):

Q1 (Cantor spectrum). Does there exist a finitely generated group $\Gamma$ and a finite symmetric generating set $S$ such that the spectrum of the adjacency operator of the Cayley graph ${\rm Cay}(\Gamma,S)$, acting on $\ell^2(\Gamma)$, is a Cantor set, i.e. a non-empty, perfect, nowhere dense compact subset of $\mathbb{R}$?

Q2 (torsion-free multi-band spectrum). Does there exist a torsion-free finitely generated group $\Gamma$ (no elements of finite order other than the identity) and a finite symmetric generating set $S$ such that the spectrum of the adjacency operator of ${\rm Cay}(\Gamma,S)$ is a disjoint union of $N\ge 2$ closed intervals (a pure band spectrum with at least two bands)?

An affirmative answer to Q2 necessarily produces a non-trivial projection in the reduced group C*-algebra of a torsion-free group and thus a counterexample to the Kadison–Kaplansky conjecture, while a general negative answer would follow from an affirmative resolution of that conjecture.

References

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