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Israel Journal of Mathematics

, Volume 81, Issue 1–2, pp 65–96 | Cite as

On the spectrum of the sum of generators for a finitely generated group

  • Pierre de la Harpe
  • A. Guyan Robertson
  • Alain Valette
Article

Abstract

Let Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π.

For the universal representationπ un of Γ, we prove in particular the following results. First, the spectrum Sp(π un (h)) contains the complex numberz of modulus one iff Sp(π un (h)) is invariant under multiplication byz, iff there exists a character\(\chi :\Gamma \to \mathbb{T}\) such that η(S)={z}. Second, forS −1=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(π un (h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.

Keywords

Hilbert Space Irreducible Representation Unitary Representation Cayley Graph Free Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University 1993

Authors and Affiliations

  • Pierre de la Harpe
    • 1
  • A. Guyan Robertson
    • 2
  • Alain Valette
    • 3
  1. 1.Section de mathématiquesGenève 24Switzerland
  2. 2.Mathematics DepartmentUniversity of NewcastleAustralia
  3. 3.Institut de mathématiquesNeuchâtelSwitzerland

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