On the question 'Can one hear the shape of a group?' and a Hulanicki type theorem for graphs
| Autor: | Rostislav Grigorchuk, Artem Dudko |
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| Rok vydání: | 2020 |
| Předmět: |
Group (mathematics)
General Mathematics 010102 general mathematics Spectrum (functional analysis) 0102 computer and information sciences Type (model theory) 01 natural sciences Omega Combinatorics 010201 computation theory & mathematics Solvable group Finitely generated group Continuum (set theory) 0101 mathematics Equivalence (measure theory) Mathematics |
| Zdroj: | Israel Journal of Mathematics |
| ISSN: | 0021-2172 |
| DOI: | 10.1007/s11856-020-1994-z |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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