Key subgroups in topological groups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909331897712640 |
|---|---|
| author | Megrelishvili, Michael Shlossberg, Menachem |
| author_facet | Megrelishvili, Michael Shlossberg, Menachem |
| contents | We introduce two minimality properties of subgroups in topological groups. A subgroup $H$ is a key subgroup (co-key subgroup) of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$ (resp., on the coset space $G/H$) the original topology. Every co-minimal subgroup is a key subgroup while the converse is not true. Every locally compact co-compact subgroup is a key subgroup (but not always co-minimal). Any relatively minimal subgroup is a co-key subgroup (but not vice versa). Extending some results concerning the generalized Heisenberg groups, we prove that the center ("corner" subgroup) of the upper unitriangular group $\mathrm{UT(n,K)}$, defined over a commutative topological unital ring $K$, is a key subgroup. Every "non-corner" 1-parameter subgroup $H$ of $\mathrm{UT(n,K)}$ is a co-key subgroup. We study injectivity property of the restriction map $$r_H \colon \mathcal{T}_{\downarrow}(G) \to \mathcal{T}_{\downarrow}(H), \ σ\mapsto σ|_H$$ and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup $H$, where $\mathcal{T}_{\downarrow}(G)$ is the semilattice of coarser Hausdorff group topologies on $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06785 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Key subgroups in topological groups Megrelishvili, Michael Shlossberg, Menachem General Topology Group Theory 20H20, 20G25, 54H11, 54H13 We introduce two minimality properties of subgroups in topological groups. A subgroup $H$ is a key subgroup (co-key subgroup) of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$ (resp., on the coset space $G/H$) the original topology. Every co-minimal subgroup is a key subgroup while the converse is not true. Every locally compact co-compact subgroup is a key subgroup (but not always co-minimal). Any relatively minimal subgroup is a co-key subgroup (but not vice versa). Extending some results concerning the generalized Heisenberg groups, we prove that the center ("corner" subgroup) of the upper unitriangular group $\mathrm{UT(n,K)}$, defined over a commutative topological unital ring $K$, is a key subgroup. Every "non-corner" 1-parameter subgroup $H$ of $\mathrm{UT(n,K)}$ is a co-key subgroup. We study injectivity property of the restriction map $$r_H \colon \mathcal{T}_{\downarrow}(G) \to \mathcal{T}_{\downarrow}(H), \ σ\mapsto σ|_H$$ and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup $H$, where $\mathcal{T}_{\downarrow}(G)$ is the semilattice of coarser Hausdorff group topologies on $G$. |
| title | Key subgroups in topological groups |
| topic | General Topology Group Theory 20H20, 20G25, 54H11, 54H13 |
| url | https://arxiv.org/abs/2309.06785 |