Group representations and construction of minimal topological groups

We show that every Hausdorff topological group is a group retract of a minimal topological group. This first was conjectured by Pestov in 1983. Our main result leads to a solution of some problems of Arhangel'skii. One of them is the problem about representability of a group as a quotient of a minimal group (Problem 519 in the first edition of "Open Problems in Topology"). Our approach is based on generalized Heisenberg groups and on groups arising from group representations on Banach spaces and in bilinear mappings.

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Every topological group is a group retract of a minimal group

Megrelishvili¹

2008

“…Indeed, let H be a non-greversible locally compact metric abelian group from Corollary 4.5. It was proved in [10] that H is a group retract of a locally compact metric minimal (thus, g-reversible) group G. In particular, H is closed in G. What is the answer if G is assumed to be even reversible?…”

Section: G-reversibility In Closed Subgroupsmentioning

confidence: 99%

Topological groups all continuous automorphisms of which are open

Chatyrko

Shakhmatov

2020

A topological space is reversible if each continuous bijection of it onto itself is open. We introduce an analogue of this notion in the category of topological groups: A topological group G is g-reversible if every continuous automorphism of G (=continuous isomorphism of G onto itself) is open. The class of g-reversible groups contains Polish groups, locally compact σ-compact groups, minimal groups, abelian groups with the Bohr topology, and reversible topological groups. We prove that subgroups of R n are g-reversible, for every positive integer n. An example of a compact (so reversible) metric abelian group having a countable dense non-greversible subgroup is given. We also highlight the differences between reversible spaces and g-reversible topological groups. Many open problems are scattered throughout the paper.All topological groups considered in this paper are supposed to be Hausdorff. The symbol R denotes the additive group of real numbers in its usual topology. By Q and Z we denote subgroups of R consisting of rational numbers and integer numbers, respectively (in their subspace topology). The quotient group T = R/Z is called the torus group.We refer the reader to [1], [9], [13] or [4] for necessary information on topological groups, and to [7] for undefined topological notions.

“…Then H(K) is a minimal and locally compact [37]. It is convenient to describe the group H(K) in the matrix form …”

mentioning

confidence: 99%

Cardinal invariants and convergence properties of locally minimal groups

Dikranjan

Shakhmatov

2020

If G is a locally essential subgroup of a compact abelian group K, then:Items (i)-(iii) hold when G is a dense locally minimal subgroup of K. We show that locally minimal locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007.For every uncountable cardinal κ, we construct a Fréchet-Urysohn minimal group G of character κ such that the connected component of G is an open normal ω-bounded subgroup (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.All topological groups are assumed to be Hausdorff.