We discuss some results about derivations and crossed homomorphisms arising in the context of locally compact groups and their group algebras, in particular, L¹(G), the von Neumann algebra VN(G) and actions of G on related algebras. We answer a question of Dales, Ghahramani, Grønbæk, showing that L¹(G) is always permanently weakly amenable. Then we show that for some classes of groups (e.g. IN-groups) the homology of L¹(G) with coefficients in VN(G) is trivial. But this is no longer true, in general, if VN(G) is replaced by other von Neumann algebras, like ℬ(L²(G)). Finally, as an example of a non-discrete, non-amenable group, we investigate the case of G = SL(2,ℝ) where the situation is rather different.