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Spectral distribution and L2-isoperimetric profile of Laplace operators on groups

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Abstract

We give a formula relating the L 2-isoperimetric profile to the spectral distribution of a Laplace operator on a finitely generated group Γ. We prove the asymptotic stability of the spectral distribution under changes of measures with finite second moment. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the Følner’s function of the group. This leads to upper bounds on spectral distributions of some non-solvable amenable groups and to sharp estimates of the spectral distributions of some solvable groups with exponential growth.

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Author information

Author notes

  1. Christophe Pittet

    Present address: Laboratoire Poncelet CNRS, Moscow, Russia

  2. Roman Sauer

    Present address: Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany

Authors and Affiliations

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

    Alexander Bendikov

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

    Christophe Pittet

  3. Department of Mathematics, University of Chicago, Chicago, USA

    Roman Sauer

Authors
  1. Alexander Bendikov
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  2. Christophe Pittet
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  3. Roman Sauer
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Correspondence to Roman Sauer.

Additional information

To the memory of Andrzej Hulanicki.

A. Bendikov was supported by the University of Aix-Marseille I as an invited Professor and by the Polish Government Scientific Research Fund, Grant NN201371736. Ch. Pittet was supported by the CNRS and the Marie Curie Transfer of Knowledge Fellowship of the European Community’s Sixth Framework Program under contract number MTKD-CT-2004-013389 with the University of Wroclaw. A. Bendikov and Ch. Pittet are grateful to Prof. E. Damek who managed the ToK contract, and to Prof. A. Grigor’yan for an invitation at the University of Bielefeld. They are also grateful to the Erwin Schrödinger Institute for several invitations. R. Sauer was supported by DFG Grant SA 1661/1-2. All authors are grateful for the financial support from Prof. W. Lück’s Leibniz award for a meeting at the WWU Münster.

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Bendikov, A., Pittet, C. & Sauer, R. Spectral distribution and L2-isoperimetric profile of Laplace operators on groups. Math. Ann. 354, 43–72 (2012). https://doi.org/10.1007/s00208-011-0724-6

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  • DOI: https://doi.org/10.1007/s00208-011-0724-6

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