Michael Megrelishvili

Key and co-key subgroups in topological groups.

(Based on a joint research project [4] with Menachem Shlossberg)

Let H be a subgroup of a Hausdorff topological group (G,\gamma). We say that

  1. H is a key subgroup of G if for every coarser Hausdorff group topology \gamma_1 \subseteq \gamma on G satisfying \gamma_1|_H = \gamma|_H it holds that \gamma_1 = \gamma.
  2. H is a co-key subgroup of G if for every coarser Hausdorff group topology \gamma_1 \subseteq \gamma on G satisfying \gamma_1/H=\gamma/H it holds that \gamma_1 = \gamma.

Key subgroups appear implicitly in some earlier publications [3,1,2,5]. 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 [3,1] concerning the generalized Heisenberg groups, we prove that the center (“corner” subgroup) of the upper unitriangular group \rm{UT}(n,K), defined over a commutative topological unital ring K, is a key subgroup. Every “non-corner” one-parameter subgroup H of \rm{UT}(n,K) is a co-key subgroup.

[1] D. Dikranjan, M. Megrelishvili, Relative minimality and co-minimality of subgroups in topological groups Topology Appl. 157 (2010), 62—76.

[2] D. Dikranjan, M. Megrelishvili, Minimality Conditions in Topological Groups, in: Recent Progress in General Topology III, 229—327, K.P. Hart, J. van Mill, P. Simon (Eds.), Springer, Atlantis Press, 2014.

[3] M. Megrelishvili, Group representations and construction of minimal topological groups, Topology Appl. 62 (1995), 1—19

[4] M. Megrelishvili, M. Shlossberg, Key and co-key subgroups in topological groups, arXiv (2023). https://arxiv.org/abs/2309.06785v2.

[5] M. Megrelishvili, M. Shlossberg, Minimality of topological matrix groups and Fermat primes, Topology Appl. 322 (2022), 108272.