On the first homology of the group of equivariant Lipschitz homeomorphisms

We study the structure of the group of equivariant Lipschitz homeomorphisms of a smooth G -manifold M which are isotopic to the identity through equivariant Lipschitz homeomorphisms with compact support. First we show that the group is perfect when M is a smooth free G -manifold. Secondly in the case of \mathbf{C}^n with the canonical U(n) -action, we show that the first homology group admits continuous moduli. Thirdly we apply the result to the case of the group L(\mathbf{C},0) of Lipschitz homeomorphisms of \mathbf{C}^n fixing the origin.