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

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 $G_omega$, $omegain{0,1,2}^mathbb N$, constructed by the second author in 1984. We show that among them there is a continuum of pairwise non-quasi-isometric groups with $mathrm{sp}(G_omega)=[- frac{1}{2},0]cup[ frac{1}{2},1]$. 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.

published proceedings

  • ISRAEL JOURNAL OF MATHEMATICS

author list (cited authors)

  • Dudko, A., & Grigorchuk, R.

citation count

  • 6

complete list of authors

  • Dudko, Artem||Grigorchuk, Rostislav

publication date

  • June 2020