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 $C^n$with the canonical $U\left(n\right)$-action, we show that the first homology group admits continuous moduli. Thirdly we apply the result to the case of the group $L(C,0)$ of Lipschitz homeomorphisms of $C$ fixing the origin.