Hausdorffness and vanishing of the first cohomology group H1(G,M)

Anthony Narkawicz

Bekka and Valette (among others) have showed us that there are very deep connections between the group G, the Banach space M (on which G acts continuously and linearly), and topological properties of the cohomology group H1(G,M). Applications of this study have been used in electrical engineering. This paper answers the following two questions. First, what does the vanishing of the group H1(G,M) imply about G? Secondly, what are necessary and sufficient conditions on G and M so that H1(G,M) is a Hausdorff space? The results are as follows. If H1(G,M) vanishes and M is contained in Lp(G) for some p, then G has exactly 1 end. Also, H1(G,M) is not Hausdorff if and only if there is a sequence fi in M with norm 1 (||fi|| = 1) for all i such that ||(g-1)fi|| --> 0 as i --> infinity for every g in G. These results give useful tools for studying the dimension of groups of the form H1(G,M). Specifically, we attempt to examine the groups H1(Zn,C*red(Zn)). The relationship between the C*-algebra C*red(G) and a specific algebra of continuous functions allows us to examine these groups very closely.