Topological Group Research Articles (Page 1)

Topologically independent sets in topological groups and vector spaces

Normality and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">N</mml:mi></mml:math>-factorizable topological groups

This article serves as a continuation of our previous work 1, which remains our primary reference for investigating specific homological properties with completion. Let the rings not be necessarily commutative and the modules be the unitary left (resp. right) modules. Let (G, (Gn) N element of N) be a filtered normal group equipped with the group topology associated with the filtration (Gn)nEN formed of normal subgroups and C(G) the set of Cauchy sequences with values in G. We define an equivalence relation R on C(G) by: (xn)R(yn) (xn)-(yn)= (xn-yn) converges to 0, noted by (xn-yn) 0. The quotient set C(G)/R:+{(xn)| (xn)element of C(G)} denoted G is equipped with a group structure and is called the completed groupe of G. For any filtered ring (resp. left A-module) (A, (In)nelement ofN (resp.(M,(Mn)nelement of N), the completed group A (resp. M) is equipped with a ring structure (resp. A-module) by (an ) x (bn ) = (anbn ) (resp.(an).(mn)=(an.mn)) where (an), (bn) element of A(resp. (mn ) element of M) called completed ring (resp. module) of A (resp. M). And for all saturated multiplicative subset S of A that satisfies the left Ore conditions, S = {(xn) element of A ∣ (xn ) not-equal 0 and ∃ n0 element of N, n greater than or equal to n0,xn element of S} is a saturated multiplicative subset of A that satisfies the left Ore conditions 1. Among the main results of this article, we have : – the functors S-1 () is isomorphic toS-1(A) tensor product A−. and S-1 () is isomorphic to S-1(A) tensor productA−. – the functors Home A(S-1A tensor productAM,−) and HOM A(S-1Atensor productA M,−) are isomorphic. – the functors S-1Atensor productA – and HomA(S-1A,−) are adjoints. This Study Allows How Establish a Relationship Between Completion [2] and Localization [4] Under the Assumptions of a Topological Structure.

This work establishes definitive conditions for the inheritance of categorical completeness and cocompleteness by categories of internal group objects. We prove that while the completeness of Grp(C) follows unconditionally from the completeness of the base category C, cocompleteness requires C to be regular, cocomplete, and admit a free group functor left adjoint to the forgetful functor. Explicit limit and colimit constructions are provided, with colimits realized via coequalizers of relations induced by group axioms over free group objects. Applications demonstrate cocompleteness in topological groups, ordered groups, and group sheaves, while Lie groups serve as counterexamples revealing necessary analytic constraints—particularly the impossibility of equipping free groups on non-discrete manifolds with smooth structures. Further results include the inheritance of regularity when the free group functor preserves finite products, the existence of internal hom-objects in locally Cartesian closed settings, monadicity for locally presentable C, and homotopical extensions where model structures on Grp(M) reflect those of M. This framework unifies classical category theory with geometric obstruction theory, resolving fundamental questions on exactness transfer and enabling new constructions in homotopical algebra and internal representation theory.

Introduction to the proceedings of the Workshop on Topology and Topological Groups (WTTG2023) held at the African Institute for Mathematical Sciences (AIMS) in Muizenberg (South Africa) on the 16th and 17th of November 2023

In this article, firstly, we introduce the concepts of weak q-spaces and weak sq-spaces. Some properties of these spaces are discussed. Secondly, we study weak q-spaces and weak sq-spaces in topological groups in terms of preimages of submetrizable spaces. We show that a topological group G is a weak q-space (resp., weak sq-space) if and only if it is an open countable-compact (resp., sequential-compact) preimage of a submetrizable space.Finally, we give some characterizations of weakly feathered, weak q-spaces and weak sq-spaces in topological groups in coset spaces as follows:(1) Let H be a closed neutral subgroup of a topological group G. Then G/H is weakly feathered if and only if G/H is an open perfect preimage of a submetrizable space.(2) Let H be a closed neutral subgroup of a topological group G. Then G/H is a weak q-space (resp., weak sq-space) if and only if G/H is an open countable-compact (resp., sequential-compact) preimage of a submetrizable space.

We study spaces X for which the space Homp(X) of automorphisms with the topology of point-wise convergence is a topological group. We identify large classes of spaces X for which Homp(X) is or is not a topological group.

Topological groups under the looking glass of preradicals

Uniform structure and topological group properties of non-additive measure algebras

In the paper we apply free products of topological groups for investigating the $M$-equivalence of Tychonoff spaces which are infinite disjoint sums of its subspaces. The main result is the following:Let $X=\overset{\infty}{\underset{i=1}{\oplus}}X_{i}$ and for every $i\in \mathbb N$ there exists a topological group $G_{i}$ such that $F(X_{i+1})$ is topologically isomorphic to the free product $F(X_{i})$ and $G_{i}$.Let $\{ n_{k} \}_{k=1}^{\infty}$ be an~increasing sequence of the natural numbers and let $\tilde{X}=\overset{\infty}{\underset{k=1}{\oplus}}X_{n_{k}}$. Then $X\overset{M}{\sim }\tilde{X}$.This theorem give us many examples of topological spaces with topologically isomorphic free topological groups. We use obtained results for constructing M-equivalent pairs and M-equivalent bundles of such spaces.

We present a general result about generating group topologies by pseudo-norms. Namely, we show that if a topology has a base of sets which are closed in a certain sense, then it can be generated by a collection of pseudo-norms such that the balls in these pseudo-norms are also closed in the same sense. The examples include linear and locally convex topologies on vector spaces, locally solid and Fatou topologies on vector lattices and Fréchet-Nikodým topologies on Boolean algebras.

The topology and symmetry group of a free boundary minimal surface in the three-dimensional Euclidean unit ball do not determine the surface uniquely. We provide pairs of non-isometric free boundary minimal surfaces having any sufficiently large genus g g , three boundary components and antiprismatic symmetry group of order 4 ( g + 1 ) 4(g+1) .

This paper presents a comparative study between the soft topological group, which was introduced in [E. H. Hamouda, On soft topological groups and soft function spaces, New Mathematics and Natural Computation, doi:10.1142/S1793005726500109], and the ordinary topological group. We explore the extension and characterization of several fundamental properties from the theory of topological groups to that of soft topological groups. The study includes establishing soft continuity of the group operation and inverse mapping, as well as examining topological concepts such as interior and closure within an algebraic framework. Furthermore, we demonstrate that the soft product of soft topological groups forms a soft topological group. The paper also investigates the concepts of soft groups and soft subspaces within this context.

The homeomorphism group of a topological surface \(\Sigma\), Homeo(\(\Sigma\)), admits a topology known as the compact-open topology, with which it becomes a topological group. In this work, we provide a self-contained proof of this fact. Moreover, we use elementary tools to prove that Homeo(\(\Sigma\)) is a Polish group (i.e., it is separable and completely metrizable). We translate the isotopy relation in Homeo(\(\Sigma\)) as path-connectedness in Homeo(\(\Sigma\)) and, denoting by Homeo0(\(\Sigma\)) the identity path component, we use classic results in Descriptive Set Theory to prove that the Extended Mapping Class Group of \(\Sigma\), Mod±(\(\Sigma\)) := Homeo(\(\Sigma\))/ Homeo0(\(\Sigma\)), is a Polish group with the quotient topology. At the end of this survey, we discuss an alternative proof of this result based on realizing the Extended Mapping Class Group as the automorphism group of the complex of curves; this connection arises as one of the most important and beautiful in the theory of Mapping Class Groups.

Let G be a group and σ,τ be topological group topologies on G. We say that σ is a successor of τ if σ is strictly finer than τ and there is no group topology properly between them. In this note, we explore the existence of successor topologies in topological groups, particularly focusing on non-abelian connected locally compact groups. Our main contributions are twofold: for a connected locally compact group (G,τ), we show that (1) if (G,τ) is compact, then τ has a precompact successor if and only if there exists a discontinuous homomorphism from G into a simple connected compact group with dense image, and (2) if G is solvable, then τ has no successors. The result (1) implies that the topology on a connected compact Lie group does not have a successor. Our work relies on the previous characterization of locally compact group topologies on abelian groups having successors.

Countable dense homogeneity and topological groups

Abstract In this paper, we provide a comprehensive classification of Stein's groups, which generalize the well‐known Higman–Thompson groups. Stein's groups are defined as groups of piecewise linear bijections of an interval with finitely many breakpoints and slopes belonging to specified additive and multiplicative subgroups of the real numbers. Our main result establishes a classification theorem for these groups under the assumptions that the slope group is finitely generated and the additive group has rank at least 2. We achieve this by interpreting Stein's groups as topological full groups of ample groupoids. A central concept in our analysis is the notion of ‐rigidity in the cohomology of groupoids. In the case where the rank of the additive group is 1, we adopt a different approach using attracting elements to impose strong constraints on the classification.

If G is a finitely generated group and X a G -set, the growth of the action of G on X is the function that measures the largest cardinality of a ball of radius n in the (possibly non-connected) Schreier graph \Gamma(G,X) . We consider the following stability problem: if G,H are finitely generated groups admitting a faithful action of growth bounded above by a function f , does the free product G \ast H also admit a faithful action of growth bounded above by f ? We show that the answer is positive under additional assumptions, and negative in general. In the negative direction, our counter-examples are obtained with G either the commutator subgroup of the topological full group of a minimal and expansive homeomorphism of the Cantor space, or G a Houghton group. In both cases, the group G admits a faithful action of linear growth, and we show that G\ast H admits no faithful action of subquadratic growth provided H is non-trivial. In the positive direction, we describe a class of groups that admit actions of linear growth and is closed under free products and exhibit examples within this class, among which the Grigorchuk group.

Given a topologically free action of a countably infinite amenable group on the Cantor set, we prove that, for every subgroup G of the topological full group containing the alternating group, the group von Neumann algebra \mathscr{L}G is a McDuff factor. This yields the first examples of nonamenable simple finitely generated groups G for which \mathscr{L}G is McDuff. Using the same construction we show moreover that if a faithful action G\curvearrowright X of a countable group on a countable set with no finite orbits is amenable then the crossed product of the associated shift action over a given II _{1} factor is a McDuff factor. In particular, if H is a nontrivial countable ICC group and G\curvearrowright X is a faithful amenable action of a countable ICC group on a countable set with no finite orbits, then the group von Neumann algebra of the generalized wreath product H\wr_{X} G is a McDuff factor. Our technique can also be applied to show that if H is a nontrivial countable group and G\curvearrowright X is an amenable action of a countable group on a countable set with no finite orbits, then the generalized wreath product H\wr_{X} G is Jones–Schmidt stable.

Two observations on extremally disconnected topological groups