Group Cohomology, Harmonic Functions and the First L ² -Betti Number

Abstract

For an infinite, finitely generated group Γ, we study the first cohomology group H ¹(Γ,λ^Γ) with coefficients in the left regular representation λ^Γ of Γ on ℓ²(Γ). We first prove thatH Γ(Γ, C Γ) embeds into HΓ(Γ,λ^Γ); as a consequence, ifH Γ(Γ,λ^Γ)=0, then Γ is not amenable with one end. For a Cayley graph X of Γ, denote by HD(X) the space of harmonic functions on X with finite Dirichlet sum. We show that, if Γ is not amenable, then there is a natural isomorphism betweenH Γ(Γ,λ^Γ) and $HD(X)/\mathbb{C}$ (the latter space being isomorphic to the first Lℓ-cohomology space of Γ). We draw the following consequences:

(1) If Γ has infinitely many ends, then $HD(X) \ne \mathbb{C}$;

(2) If Γ has Kazhdan's property (T)>, then $HD(X) = \mathbb{C}$;

(3) The property H ¹(Γ, λ^Γ)=0 is a quasi-isometry invariant;

(4) Either HΓ(Γ,λ^Γ) or HΓ(Γ,λ^Γ) is infinite-dimensional;

(5) If $\Gamma = \Gamma _1 \times \Gamma _2$ with $\Gamma <Superscript>1</Superscript>$ non-amenable and $\Gamma <Superscript>2</Superscript>$ infinite, then HΓ(Γ,λ^Γ)=0