Groups not acting on compact metric spaces by homeomorphisms

TL;DR

This paper proves that an uncountable direct sum of non-Abelian groups cannot be embedded into the group of homeomorphisms of any compact metric space, highlighting limitations on group actions.

Contribution

It establishes a new non-embedding result for large non-Abelian groups acting on compact metric spaces, extending understanding of group actions.

Findings

Uncountably many non-Abelian groups do not embed into homeomorphism groups

Limits on group actions on compact metric spaces

Advances in understanding group action restrictions

Abstract

We show that the direct sum of uncountably many non-Abelian groups does not embed into the group of homeomorphisms of a compact metric space.