First l^2-Cohomology Groups

Abstract

We want to take a look at the first cohomology group H^1(G, l^2(G)), in particular when G is locally-finite. First, though, we discuss some results about the space H^1(G, C G) for G locally-finite, as well as the space H^1(G, l^2(G)) when G is finitely generated. We show that, although in the case when G is finitely generated the embedding of C G into l^2(G) induces an embedding of the cohomology groups H^1(G, C G) into H^1(G, l^2(G)), when G is countably-infinite locally-finite, the induced homomorphism is not an embedding. However, even though the induced homomorphism is not an embedding, we still have that H^1(G, l^2(G)) neq 0 when G is countably-infinite locally-finite. Finally, we give some sufficient conditions for H^1(G,l^2(G)) to be zero or non-zero.