Hausdorffness and Vanishing of the First Cohomology Group H(g,m)

This paper characterizes the first cohomology group H(G,M) where M is a Banach space (with norm || ||M) that is also a left CG module such that the elements of G act on M as continuous C-linear transformations. We study this group for G an infinite, finitely generated group. Of particular interest are the implications of the vanishing of the group H(G,M). The first result is that H(G,CG) imbeds in H(G,M) whenever CG ⊂ M ⊂ L(G) for some p ∈ N. This is an unpublished result and shows immediately that if H(G,M) = 0, then G can have only 1 end. Secondly (also a new result), we show that H(G,M) is not Hausdorff if and only if there exist fi ∈ M with norm 1 (||fi||M = 1) for all i with the property that ||gfi − fi||M −→ 0 as i −→ ∞ for every g ∈ G. This is then used to show that if M and || ||M satisfy certain properties and if G satisfies a “strong Følner condition,” then H(G,M) is not Hausdorff. For the second half of this paper, we give several applications of these last two theorems focusing on the group G = Z. These applications were the main source of motivation for this project.