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.
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de la Harpe, P., Robertson, A.G. & Valette, A. On the spectrum of the sum of generators for a finitely generated group. Israel J. Math. 81, 65–96 (1993). https://doi.org/10.1007/BF02761298
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DOI: https://doi.org/10.1007/BF02761298