Key subgroups in topological groups

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>H</m:mi> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>UT</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">n</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{UT(n,K)}} , defined over a commutative topological unital ring K , is a key subgroup. Every “non-corner” 1-parameter subgroup H of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>UT</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">n</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{UT(n,K)}} is a co-key subgroup. We study injectivity property of the restriction map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>r</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒯</m:mi> <m:mo>↓</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒯</m:mi> <m:mo>↓</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>H</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {r_{H}:\mathcal{T}_{\downarrow}(G)\to\mathcal{T}_{\downarrow}(H)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>σ</m:mi> <m:mo>↦</m:mo> <m:msub> <m:mrow> <m:mi>σ</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>H</m:mi> </m:msub> </m:mrow> </m:math> {\sigma\mapsto\sigma|_{H}} and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup H , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒯</m:mi> <m:mo>↓</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{T}_{\downarrow}(G)} is the semilattice of coarser Hausdorff group topologies on G .