Non-Roelcke precompactness of groups of surface homeomorphisms

TL;DR

This paper demonstrates that certain subgroups of surface homeomorphisms with highly transitive actions cannot be Roelcke precompact under the compact-open topology.

Contribution

It establishes a new non-compactness result for subgroups of surface homeomorphisms with transitive actions, expanding understanding of their topological properties.

Findings

Subgroups with sufficiently transitive actions are not Roelcke precompact.

The result applies to boundary-fixing homeomorphisms of compact surfaces.

It advances the theory of topological groups of surface homeomorphisms.

Abstract

We prove that no subgroup of the group of boundary-fixing homeomorphisms of a compact surface whose action on the interior of the surface is sufficiently transitive can be Roelcke precompact with the topology inherited from the compact-open topology.