Can one hear the shape of a group?

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Let Γ be a finitely generated group, and letS be a finite, non-necessarily symmetric, generating subset of Γ. Leth be the transition operator of the directed Cayley graphG(Γ,S), acting onl 2 (Γ). Staring with Kesten’s seminal results, we give a survey of results linking group-theoretic properties of the pair (Γ,S) with spectral properties ofh.