Successors of topologies of connected locally compact groups

2024/07/19 by Peng, Dekui, Xiao, Zhiqiang
#22A05 #22C05 #22D05 #54A10 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2407.14323

Abstract

Let G be a group and σ, τ be topological group topologies on G. We say that σ is a successor of τ if σ is strictly finer than τ 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, τ), we show that (1) if (G, τ) is compact, then τ 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 τ has no successors. Our work relies on the previous characterization of locally compact group topologies on abelian groups processing successors.

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