On the question "Can one hear the shape of a group?" and Hulanicki type theorem for graphs

TL;DR

This paper demonstrates that the spectrum of a finitely generated group does not uniquely determine the group, providing examples of non-quasi-isometric groups with identical spectra and extending Hulanicki's theorem to graphs.

Contribution

It shows the spectrum cannot distinguish certain groups and extends Hulanicki's theorem to graph spectra, with explicit examples of groups having identical spectra.

Findings

Existence of non-quasi-isometric groups with identical spectra

Construction of continuum many torsion-free solvable groups sharing the same spectrum

Extension of Hulanicki's theorem to graph spectra

Abstract

We study the question of whether it is possible to determine a finitely generated group $G$ up to some notion of equivalence from the spectrum $\mathrm{sp}(G)$ of $G$. We show that the answer is "No" in a strong sense. As the first example we present the collection of amenable 4-generated groups $G_\omega$, $\omega\in\{0,1,2\}^\mathbb N$, constructed by the second author in 1984. We show that among them there is a continuum of pairwise non-quasi-isometric groups with $\mathrm{sp}(G_\omega)=[-\tfrac{1}{2},0]\cup[\tfrac{1}{2},1]$. Moreover, for each of these groups $G_\omega$ there is a continuum of covering groups $G$ with the same spectrum. As the second example we construct a continuum of $2$-generated torsion-free step-3 solvable groups with the spectrum $[-1,1]$. In addition, in relation to the above results we prove a version of Hulanicki Theorem about inclusion of spectra for covering graphs.