On the boundedness of equivariant homeomorphism groups

Abstract

Given a principal $G$-bundle $\pi:M\to B$, let $\mathcal{H}_G(M)$ be the identity component of the group of $G$-equivariant homeomorphisms on $M$. The problem of the uniform perfectness and boundedness of $\mathcal{H}_G(M)$ is studied. It occurs that these properties depend on the structure of $\mathcal{H}(B)$, the identity component of the group of homeomorphisms of $B$, and of $B$ itself. Most of the obtained results still hold in the $C^r$ category.