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

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 $G_omega$, $omegain{0,1,2}^mathbb N$, 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.