Successors of topologies of connected locally compact groups

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.