Successors of topologies of connected locally compact groups
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916330361323520 |
|---|---|
| author | Peng, Dekui Xiao, Zhiqiang |
| author_facet | Peng, Dekui Xiao, Zhiqiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14323 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Successors of topologies of connected locally compact groups Peng, Dekui Xiao, Zhiqiang Group Theory General Topology 22A05, 54A10, 22D05, 22C05 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. |
| title | Successors of topologies of connected locally compact groups |
| topic | Group Theory General Topology 22A05, 54A10, 22D05, 22C05 |
| url | https://arxiv.org/abs/2407.14323 |