ISSN:
1572-929X
Keywords:
Group cohomology
;
ends
;
harmonic functions
;
L 2-cohomology.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract For an infinite, finitely generated group Γ, we study the first cohomology group H 1(Γ,λΓ) with coefficients in the left regular representation λΓ of Γ on ℓ2(Γ). We first prove thatH Γ(Γ, C Γ) embeds into HΓ(Γ,λΓ); as a consequence, ifH Γ(Γ,λΓ)=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,λΓ) 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, λΓ)=0 is a quasi-isometry invariant; (4) Either HΓ(Γ,λΓ) or HΓ(Γ,λΓ) 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,λΓ)=0
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1017974406074
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