Topical Collection "Topological Groups"

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 March 31, 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.

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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

Published Papers (13 papers)

2017

Jump to: 2016, 2015

Open AccessArticle Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group
Axioms 2017, 6(4), 27; doi:10.3390/axioms6040027
Received: 28 August 2017 / Revised: 9 October 2017 / Accepted: 9 October 2017 / Published: 20 October 2017
PDF Full-text (299 KB)
Abstract
The scale of an endomorphism of a totally disconnected, locally compact group 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.
[...] Read more.
The scale of an endomorphism of a totally disconnected, locally compact group 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
Open AccessArticle Categorically Closed Topological Groups
Axioms 2017, 6(3), 23; doi:10.3390/axioms6030023
Received: 30 June 2017 / Revised: 26 July 2017 / Accepted: 27 July 2017 / Published: 30 July 2017
PDF Full-text (433 KB) | HTML Full-text | XML Full-text
Abstract
Let C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category C is called C -closed if for each morphism Φ X × Y in the category C the image Φ ( X ) = { y Y : x X ( x , y ) Φ } is closed in Y. In the paper we survey existing and new results on topological groups, which are C -closed for various categories C of topologized semigroups. Full article
Open AccessArticle No Uncountable Polish Group Can be a Right-Angled Artin Group
Axioms 2017, 6(2), 13; doi:10.3390/axioms6020013
Received: 28 March 2017 / Revised: 20 April 2017 / Accepted: 4 May 2017 / Published: 11 May 2017
PDF Full-text (220 KB) | HTML Full-text | XML Full-text
Abstract
We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0<k<ω, then lg(x)lg(xk
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We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in 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

2016

Jump to: 2017, 2015

Open AccessEditorial An Overview of Topological Groups: Yesterday, Today, Tomorrow
Axioms 2016, 5(2), 11; doi:10.3390/axioms5020011
Received: 18 April 2016 / Accepted: 20 April 2016 / Published: 5 May 2016
PDF Full-text (220 KB) | HTML Full-text | XML Full-text
Open AccessArticle Non-Abelian Pseudocompact Groups
Axioms 2016, 5(1), 2; doi:10.3390/axioms5010002
Received: 29 September 2015 / Revised: 24 November 2015 / Accepted: 23 December 2015 / Published: 12 January 2016
PDF Full-text (296 KB) | HTML Full-text | XML Full-text
Abstract
Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22|K| -many strictly finer pseudocompact topological group refinements.
[...] Read more.
Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22|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

2015

Jump to: 2017, 2016

Open AccessArticle Free Boolean Topological Groups
Axioms 2015, 4(4), 492-517; doi:10.3390/axioms4040492
Received: 30 August 2015 / Revised: 17 October 2015 / Accepted: 23 October 2015 / Published: 3 November 2015
PDF Full-text (300 KB) | HTML Full-text | XML Full-text
Abstract
Known and new results on free Boolean topological groups are collected. An account of the properties that these groups share with free or free Abelian topological groups and properties specific to free Boolean groups is given. Special emphasis is placed on the application
[...] Read more.
Known and new results on free Boolean topological groups are collected. An account of the properties that these groups share with free or free Abelian topological groups and properties specific to free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups. Full article
Open AccessArticle Characterized Subgroups of Topological Abelian Groups
Axioms 2015, 4(4), 459-491; doi:10.3390/axioms4040459
Received: 2 September 2015 / Revised: 27 September 2015 / Accepted: 8 October 2015 / Published: 16 October 2015
Cited by 2 | PDF Full-text (352 KB) | HTML Full-text | XML Full-text
Abstract
A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {xX : vn(x) → 0 in T}.
[...] Read more.
A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {xX : vn(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
Open AccessArticle Locally Quasi-Convex Compatible Topologies on a Topological Group
Axioms 2015, 4(4), 436-458; doi:10.3390/axioms4040436
Received: 4 May 2015 / Revised: 28 September 2015 / Accepted: 8 October 2015 / Published: 13 October 2015
Cited by 4 | PDF Full-text (336 KB) | HTML Full-text | XML Full-text
Abstract
For a locally quasi-convex topological abelian group (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.
[...] Read more.
For a locally quasi-convex topological abelian group (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
Open AccessArticle Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds
Axioms 2015, 4(3), 313-320; doi:10.3390/axioms4030313
Received: 30 April 2015 / Revised: 3 July 2015 / Accepted: 9 July 2015 / Published: 27 July 2015
Cited by 1 | PDF Full-text (192 KB) | HTML Full-text | XML Full-text
Abstract
I discuss old and new results on fixed points of local actions by Lie groups 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
Open AccessArticle Pro-Lie Groups: A Survey with Open Problems
Axioms 2015, 4(3), 294-312; doi:10.3390/axioms4030294
Received: 17 June 2015 / Revised: 15 July 2015 / Accepted: 17 July 2015 / Published: 24 July 2015
Cited by 4 | PDF Full-text (263 KB) | HTML Full-text | XML Full-text
Abstract
A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete
[...] Read more.
A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups. Full article
Open AccessArticle Lindelöf Σ-Spaces and R-Factorizable Paratopological Groups
Axioms 2015, 4(3), 254-267; doi:10.3390/axioms4030254
Received: 17 April 2015 / Revised: 25 June 2015 / Accepted: 30 June 2015 / Published: 10 July 2015
Cited by 1 | PDF Full-text (235 KB) | HTML Full-text | XML Full-text
Abstract
We prove that if a paratopological group 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 celw
[...] Read more.
We prove that if a paratopological group 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 celw(G) ≤ w, and the Hewitt–Nachbin completion of G is again an R-factorizable paratopological group. Full article
Open AccessArticle On T-Characterized Subgroups of Compact Abelian Groups
Axioms 2015, 4(2), 194-212; doi:10.3390/axioms4020194
Received: 16 February 2015 / Revised: 11 June 2015 / Accepted: 16 June 2015 / Published: 19 June 2015
Cited by 3 | PDF Full-text (283 KB) | HTML Full-text | XML Full-text
Abstract
A sequence \(\{ u_n \}_{n\in \omega}\) in abstract additively-written Abelian group \(G\) is called a \(T\)-sequence if there is a Hausdorff group topology on \(G\) relative to which \(\lim_n u_n =0\). We say that a subgroup \(H\) of an infinite compact Abelian group
[...] Read more.
A sequence \(\{ u_n \}_{n\in \omega}\) in abstract additively-written Abelian group \(G\) is called a \(T\)-sequence if there is a Hausdorff group topology on \(G\) relative to which \(\lim_n u_n =0\). We say that a subgroup \(H\) of an infinite compact Abelian group \(X\) is \(T\)-characterized if there is a \(T\)-sequence \(\mathbf{u} =\{ u_n \}\) in the dual group of \(X\), such that \(H=\{ x\in X: \; (u_n, x)\to 1 \}\). We show that a closed subgroup \(H\) of \(X\) is \(T\)-characterized if and only if \(H\) is a \(G_\delta\)-subgroup of \(X\) and the annihilator of \(H\) admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group \(X\) are \(T\)-characterized if and only if \(X\) is metrizable and connected. We prove that every compact Abelian group \(X\) of infinite exponent has a \(T\)-characterized subgroup, which is not an \(F_{\sigma}\)-subgroup of \(X\), that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427–2442). Full article
Open AccessReview Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review
Axioms 2015, 4(1), 84-101; doi:10.3390/axioms4010084
Received: 12 December 2014 / Revised: 9 January 2015 / Accepted: 26 January 2015 / Published: 4 February 2015
PDF Full-text (267 KB) | HTML Full-text | XML Full-text
Abstract
Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and
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Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. The paper lists several open problems and conjectures and tries to place this topic of research in the general context of hyperbolic and topological dynamics. Full article

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