**Abstract**

## Journal Menu

► Journal Menu# Topical Collection "Topological Groups"

A topical collection in *Axioms* (ISSN 2075-1680).

## Editor

Collection Editor
Prof. Dr. Sidney A. Morris
1. Faculty of Science and Technology, Federation University Australia, Victoria 3353, Australia
2. School of Engineering and Mathematical Sciences, La Trobe University, Bundoora, Victoria 3086, Australia Website | E-Mail Interests: topological groups especially locally compact groups; topology; group theory; functional analysis; universal algebra; numerical geometry; history of mathematics; information technology security; health informatics; international education; university education; online education; social media in the teaching of mathematics; stock market prediction; managing scholarly journals |

## Topical Collection Information

Dear Colleagues,

For over a century, topological groups have been an active area of research. In 1900, David Hilbert presented a seminal address to the International Congress of Mathematician, in which he formulated 23 problems that influenced a vast amount of research of the 20th century. The fifth of these problems, Hilbert5, asked whether every locally euclidean topological group admits a Lie group structure and this motivated an enormous effort on locally compact groups. It culminated in the work of Gleason, Iwasawa, Montgomery, Yamabe, and Zippin, yielding a positive answer to Hilbert5 and important structure theory of locally compact groups. Later, Jean Dieudonné quipped that Lie groups had moved to the centre of mathematics and that one cannot undertake anything without them. A modern introduction to Lie Groups is given in the book by Hilgert and Neeb. Recently there has been much interest in infinite-dimensional Lie groups including significant publications by Glöckner and Neeb, and two books by Hofmann and Morris, which demonstrated the power of Lie Theory in describing the structure of compact groups and (almost) connected pro-Lie Groups. Advances in profinite group theory are described in books by Wilson and by Ribes and Zaleskii and on locally compact totally disconnected groups in the papers of Willis and collaborators. Over some decades the Moscow school led by Arhangel’skii produced many beautiful results on free topological groups and non-locally compact groups in general. The book “Topological Groups and Related Structures” by Arhangel’skii and Tkachenko contains many results about such groups. In the 1960s Morris initiated the study of the classes of topological groups he called Varieties of Topological Groups, and several others have contributed to their theory. There has also been much research on pseudocompact groups by Comfort and his collaborators.

In this Topical Collection on Topological Groups we seek to address all areas of topological group theory and related structures. Original articles reporting recent progress and survey articles are sought. Authors are encouraged to include interesting open questions. **The deadline for submitting papers is June 30, 2018.**

Prof. Dr. Sidney A. Morris*Collection Editor*

**Manuscript Submission Information**

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the collection website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. *Axioms* is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 350 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

## Keywords

- topological groups
- compact groups
- profinite groups
- locally compact groups
- Lie groups
- actions by Lie groups
- pro-Lie groups
- almost periodic
- semitopological groups
- paratopological groups
- structure theory
- characterizing subgroups
- transformation groups
- representations
- free topological groups, free Boolean topological groups and free products
- variety of topological groups
- Hilbert’s 5th problem
- (locally) minimal topological groups
- compactness conditions in topological groups
- pseudocompact groups
- duality and reflexivity
- covering theory for topological groups
- suitable sets for topological groups
- algebraic topology and topological groups
- compact semigroups

## Related Special Issue

- Topological Groups: Yesterday, Today, Tomorrow in
*Axioms*(10 articles - displayed below)

**Abstract**

*G*is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of

*G*.

*G*is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of

*G*. Methods for computing the scale, which is a positive integer, are surveyed and illustrated by applying them in diverse cases, including when

*G*is compact; an automorphism group of a tree; Neretin’s group of almost automorphisms of a tree; and a

*p*-adic Lie group. The information required to compute the scale is reviewed from the perspective of the, as yet incomplete, general theory of totally disconnected, locally compact groups. Full article

**Abstract**

**Abstract**

*G*is a Polish group and

*A*a group admitting a system of generators whose associated length function satisfies: (i) if

*G*is a Polish group and

*A*a group admitting a system of generators whose associated length function satisfies: (i) if

*G*of size

*A*. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups. Full article

**Abstract**

**Abstract**

^{|K|}-many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 2

^{2|K|}-many strictly finer pseudocompact topological group refinements.

^{|K|}-many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 2

^{2|K| }-many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudocompact subgroup and a strictly finer pseudocompact topological group refinement. (Theorems (A), (B) and (C) become false if the non-metrizable hypothesis is omitted.) With a detailed view toward the relevant literature, the present authors ask: What happens to (A), (B), (C) and to similar known facts about pseudocompact abelian groups if the abelian hypothesis is omitted? Are the resulting statements true, false, true under certain natural additional hypotheses, etc.? Several new results responding in part to these questions are given, and several specific additional questions are posed. Full article

**Abstract**

**Abstract**

**v**= (v

_{n}) of characters of

*X*if

*H*= {

*x*∈

*X*: v

_{n}(

*x*) → 0 in T}.

**v**= (v

_{n}) of characters of

*X*if

*H*= {

*x*∈

*X*: v

_{n}(

*x*) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of auto-characterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proven to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups. Full article

**Abstract**

*G*,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (

*G*) that are compatible with (τ) (

*i.e.*, have the same dual as (

*G*,τ) ordered by inclusion.

*G*,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (

*G*) that are compatible with (τ) (

*i.e.*, have the same dual as (

*G*,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(

*G*,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (

*H*) of (

*G*) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (

*D*) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on

*D*(4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-

*σ*-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (

*G*,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9). Full article

**Abstract**

*G*on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed. Full article

**Abstract**

**Abstract**

*G*is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition,

*G*is regular, then it is totally w-narrow and satisfies cel

_{w}

*G*is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition,

*G*is regular, then it is totally w-narrow and satisfies cel

_{w}(

*G*) ≤ w, and the Hewitt–Nachbin completion of

*G*is again an R-factorizable paratopological group. Full article

*T*-Characterized Subgroups of Compact Abelian Groups

**Abstract**

*Topol. Appl.*

**2013**,

*160*, 2427–2442). Full article

**Abstract**