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

@inproceedings{Dudko2018OnTQ,
title={On the question “Can one hear the shape of a group?” and a Hulanicki type theorem for graphs},
author={A. Dudko and R. Grigorchuk},
year={2018}
}

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… Expand