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

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Abstract

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.

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Acknowledgements

We are grateful to the anonymous referee for careful reading of our paper and numerous useful comments and suggestions.

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Scuences, Sniadeckich 8, Warsaw, 00-656, Poland

    Artem Dudko

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Rostislav Grigorchuk

Authors
  1. Artem Dudko
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  2. Rostislav Grigorchuk
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Corresponding author

Correspondence to Artem Dudko.

Additional information

Artem Dudko acknowledges the support by the National Science Centre, Poland, grant 2016/23/P/ST1/04088 under POLONEZ programme which has received funding from the EU Horizon 2020 research and innovation programme under the MSCA grant agreement No. 665778.

Rostilav Grigorchuk was partially supported by NSF grant DMS-1207699 and by Simons Foundation Collaboration Grant for Mathematicians, Award Number 527814

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Dudko, A., Grigorchuk, R. On the question “Can one hear the shape of a group?” and a Hulanicki type theorem for graphs. Isr. J. Math. 237, 53–74 (2020). https://doi.org/10.1007/s11856-020-1994-z

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  • DOI: https://doi.org/10.1007/s11856-020-1994-z

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