Successors of topologies of connected locally compact groups

TL;DR

This paper investigates the existence of successor topologies in connected locally compact groups, revealing conditions under which such successors exist or do not, especially in compact and solvable cases.

Contribution

It characterizes when successor topologies exist in connected locally compact groups, extending previous work from abelian to non-abelian groups.

Findings

Successors exist in compact groups if a discontinuous homomorphism with dense image exists.

Solvable groups have no successor topologies.

Extension of successor topology characterization to non-abelian groups.

Abstract

Let $G$ be a group and $\sigma, \tau$ be topological group topologies on $G$. We say that $\sigma$ is a successor of $\tau$ if $\sigma$ is strictly finer than $\tau$ and there is not a group topology properly between them. In this note, we explore the existence of successor topologies in topological groups, particularly focusing on non-abelian connected locally compact groups. Our main contributions are twofold: for a connected locally compact group $(G, \tau)$, we show that (1) if $(G, \tau)$ is compact, then $\tau$ has a precompact successor if and only if there exists a discontinuous homomorphism from $G$ into a simple connected compact group with dense image, and (2) if $G$ is solvable, then $\tau$ has no successors. Our work relies on the previous characterization of locally compact group topologies on abelian groups processing successors.