Year: 2025
Type: article
Abstract: 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 {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 from [M. Megrelishvili, Group representations and construction of minimal topological groups, Topology Appl. 62 1995, 1, 1–19] and [D. Dikranjan and M. Megrelishvili, Relative minimality and co-minimality of subgroups in topological groups, Topology Appl. 157 2010, 1, 62–76] concerning the generalized Heisenberg groups, we prove that the center (“corner” subgroup) of the upper unitriangular group UT ( n , K ) {\mathrm{UT(n,K)}} , defined over a commutative topological unital ring K , is a key subgroup. Every “non-corner” 1-parameter subgroup H of UT ( n , K ) {\mathrm{UT(n,K)}} is a co-key subgroup. We study injectivity property of the restriction map r H : 𝒯 ↓ ( G ) → 𝒯 ↓ ( H ) {r_{H}:\mathcal{T}_{\downarrow}(G)\to\mathcal{T}_{\downarrow}(H)} , σ ↦ σ | H {\sigma\mapsto\sigma|_{H}} and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup H , where 𝒯 ↓ ( G ) {\mathcal{T}_{\downarrow}(G)} is the semilattice of coarser Hausdorff group topologies on G .
Source: Forum Mathematicum
Institutions: Bar-Ilan University, Reichman University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1589979182, G8053090141
OpenAlex ID: W4408919026