On the first homology of the group of equivariant Lipschitz homeomorphisms

Abstract

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.