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

We study the question of whether or not it is possible to determine a finitely generated group G up to some notion of equivalence from the spectrum sp(G) of G. We show that the answer is “No” in a strong sense. As a first example we present the collection of amenable 4-generated groups Gω, ω ∈ {0, 1, 2}ℕ, constructed by the second author in 1984. We show that among them there is a continuum of pairwise non-quasi-isometric groups with $${\rm{sp}}(G_\omega)=[-\frac{1}{2},0]\cup[\frac{1}{2},1]$$. Moreover, for each of these groups Gω there is a continuum of covering groups G with the same spectrum. As a 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 the Hulanicki Theorem about inclusion of spectra for covering graphs.