Boundedness of certain automorphism groups of an open manifold

TL;DR

This paper proves that certain automorphism groups of open manifolds with finite ends are bounded and uniformly perfect, providing bounds on their commutator length diameter based on fragmentation norms.

Contribution

It introduces conditions characterizing boundedness of automorphism groups of open manifolds and estimates their commutator length diameter using fragmentation norms.

Findings

Automorphism groups with no support restrictions are bounded.

These groups are shown to be uniformly perfect.

The commutator length diameter is estimated by fragmentation norms.

Abstract

It is shown that certain diffeomorphism or homeomorphism groups with no restriction on support of an open manifold with finite number of ends are bounded. It follows that these groups are uniformly perfect. In order to characterize the boundedness several conditions on automorphism groups of an open manifold are introduced. In particular, it is shown that the commutator length diameter of the automorphism group $\mathcal D(M)$ of a portable manifold $M$ is estimated by $2fragd_{\mathcal D(M)}+2$, where $fragd_{\mathcal D(M)}$ is the diameter of $\mathcal D(M)$ in the fragmentation norm.