GROUP COHOMOLOGY, HARMONIC-FUNCTIONS AND THE FIRST L-2-BETTI NUMBER

For an infinite, finitely generated group Gamma, we study the first co homology group H-1(Gamma, lambda(Gamma)) with coefficients in the left regular representation lambda(Gamma) of Gamma on l(2)(Gamma). We firs t 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 no t 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 s how that, if Gamma is not amenable, then there is a natural isomorphis m between H-1(Gamma, lambda(Gamma)) and HD(X)/C (the latter space bein g 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, lam bda(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, la mbda(Gamma)) = 0.