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Some Spectral and Geometric Aspects of Countable Groups

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Random Walks, Boundaries and Spectra

Part of the book series: Progress in Probability ((PRPR,volume 64))

Abstract

We discuss the relationship between the isospectral profile and the spectral distribution of a Laplace operator on a countable group. In the case of locally finite countable groups, we emphasize the relevance of the metric associated to a natural Markov operator: it is an ultra-metric whose balls are optimal sets for the isospectral profile.

Mathematics Subject Classification (2000). Primary: 60B15, 20F65; Secondary: 58C40.

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References

  1. Alexander Bendikov, Barbara Bobikau, and Christophe Pittet, Spectral properties of a class of random walks on locally finite groups, preprint.

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  2. Alexander Bendikov, Christophe Pittet, and Roman Sauer, Spectral distribution and L2-isoperimetric profile of Laplace operators on groups, arXiv:0901.0271 (2009), 1–22.

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  3. Thierry Coulhon, Alexander Grigor’yan, and Christophe Pittet, A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1763–1827 (English, with English and French summaries).

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  4. Mikhail Gromov and Mikhail Shubin, von Neumann spectra near zero, Geom. Funct. Anal 1 (1991), no. 4, 375–404.

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  5. Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Wroclaw University, Wroclaw, Poland

    Alexander Bendikov & Barbara Bobikau

  2. CMI Université d’Aix-Marseille I, Marseille Cédex 3, France

    Christophe Pittet

Authors
  1. Alexander Bendikov
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  2. Barbara Bobikau
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  3. Christophe Pittet
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Corresponding author

Correspondence to Alexander Bendikov .

Editor information

Editors and Affiliations

  1. Mathematisches Inst., Universität Jena, Ernst-Abbe-Platz 2, Jena, 07737, Germany

    Daniel Lenz

  2. Inst. Mathematik C, TU Graz, Steyrergasse 30, Graz, 8010, Austria

    Florian Sobieczky

  3. Inst. Mathematik C, TU Graz, Steyrergasse 30, Graz, 8010, Austria

    Wolfgang Woess

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Bendikov, A., Bobikau, B., Pittet, C. (2011). Some Spectral and Geometric Aspects of Countable Groups. In: Lenz, D., Sobieczky, F., Woess, W. (eds) Random Walks, Boundaries and Spectra. Progress in Probability, vol 64. Springer, Basel. https://doi.org/10.1007/978-3-0346-0244-0_12

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