Valette, Alain

1997, Bekka, Bachir, Valette, Alain

For an infinite, finitely generated group Gamma, we study the first cohomology group H-1(Gamma, lambda(Gamma)) with coefficients in the left regular representation lambda(Gamma) of Gamma on l(2)(Gamma). We first prove that H-1(Gamma, C Gamma) embeds into H-1(Gamma, lambda(Gamma)); as a consequence, if H-1(Gamma, lambda(Gamma)) = 0, then Gamma is not amenable with one end. For a Cayley graph X of Gamma, denote by HD(X) the space of harmonic functions on X with finite Dirichlet sum. We show that, if Gamma is not amenable, then there is a natural isomorphism between H-1(Gamma, lambda(Gamma)) and HD(X)/C (the latter space being isomorphic to the first L-2-cohomology space of Gamma). We draw the following consequences: (1) If Gamma has infinitely many ends, then HD(X) not equal C; (2) If Gamma has Kazhdan's property (T), then HD(X) = C; (3) The property H-1(Gamma, lambda(Gamma)) = 0 is a quasi-isometry invariant; (4) Either H-1(Gamma, lambda(Gamma)) = 0 or H-1(Gamma, lambda(Gamma)) is infinite-dimensional; (5) If Gamma = Gamma(1) x Gamma(2) with Gamma(1) non-amenable and Gamma(2) infinite, then H-1(Gamma, lambda(Gamma)) = 0.

1993, De La Harpe, Pierre, Robertson, Guyan, Valette, Alain

Let GAMMA be a finitely generated group. In the group algebra C[T], form the average h of a finite set S of generators of GAMMA. Given a unitary representation pi of GAMMA, we relate spectral properties of the operator pi(h) to ProPerties Of GAMMA and pi. For the universal representation pi(un) of GAMMA, we prove in particular the following results. First, the spectrum Sp(pi(un) (h)) contains the complex number , of modulus one iff Sp(pi(un) (h)) is invariant under multiplication by z, iff there exists a character x: GAMMA --> T such that chi(S) = {z}. Second, for S-1 = S, the group GAMMA has Kazhdan's proPertY (T) if and only if 1 is isolated in Sp(pi(un) (h)); in this case, the distance between 1 and other point, of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.