Class Of Topological Groups Research Articles

Some pseudocompact-like properties in certain topological groups

We construct in ZFC a countably compact group without non-trivial convergent sequences of size 2c, answering a question of Bellini, Rodrigues and Tomita [4]. We also construct in ZFC a selectively pseudocompact group which is not countably pracompact, showing that these two properties are not equivalent in the class of topological groups. Using the same technique, we construct a group which has all powers selectively pseudocompact but is not countably pracompact, assuming the existence of a selective ultrafilter.

Amenability versus property (𝑇) for non-locally compact topological groups

For locally compact groups amenability and Kazhdan’s property ( T ) (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugation-invariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property ( T ) (T) (i.e., admits a finite Kazhdan set). If an amenable topological group with property ( T ) (T) admits a faithful uniformly continuous representation, then it is maximally almost periodic. In particular, an extremely amenable SIN group never has strong property ( T ) (T) , and an extremely amenable subgroup of unitary operators in the uniform topology is never a Kazhdan group. This leads to first examples distinguishing between property ( T ) (T) and property ( F H ) (FH) in the class of Polish groups. Disproving a 2003 conjecture by Bekka, we construct a complete, separable, minimally almost periodic topological group with property ( T ) (T) having no finite Kazhdan set. Finally, as a curiosity, we observe that the class of topological groups with property ( T ) (T) is closed under arbitrary infinite products with the usual product topology.

Correction to: A class of topological groups which do not admit normal compatible locally quasi-convex topologies

Unfortunately an erratum appears in the statement corresponding to Theorem 6.9 of the above mentioned paper.

A class of topological groups which do not admit normal compatible locally quasi-convex topologies

We first study the extent of $${{\mathbb {Z}}}_b^{{\mathbb {R}}}$$ , the product of $${\mathfrak {c}}$$ -copies of the group of integer numbers, each of them endowed with its Bohr topology. We prove that it is exactly $${\mathfrak {c}}$$ . From this fact we derive the non-normality of any locally quasi-convex group topology on $${{\mathbb {Z}}}^{{\mathbb {R}}}$$ compatible with the duality $$({ {\mathbb {Z}}}^{{{\mathbb {R}}}}, {\mathbb {T}}^{({{\mathbb {R}}})})$$ . Finally we prove that any product of $${\mathfrak {c}}$$ -many locally compact, non countably compact, separable abelian topological groups does not admit either a normal locally quasi-convex compatible topology.

On a class of topological groups

A topological group is an SNS-group if its identity element possesses a fundamental system of neighborhoods formed by normal subgroups. In this paper we prove the existence of initial SNS-topologies, from which we derive that the class of SNS-groups is closed under the formation of products and projective limits, and we prove the existence of final SNS-topologies, from which we derive that the class of SNS-groups is closed under the formation of free products and inductive limits.

An example of a topological group

In this paper three classes of topological groups are considered: the groups which, in the sense of their topological structure, are Lindelöf P-spaces, and the groups which are o-bounded or strictly o-bounded. In the sense of covering properties these classes are close to σ-compactness. We construct an example of a topological group G which is a Lindelöf P-space but is not strictly o-bounded. The example is generated by a ladder system. We answer some questions of C. Hernandez and M. Tkachenko.

Supercritical Spatially Homogeneous Branching in Admits no Equilibria

At the beginning of investigations in spatially homogeneous branching processes in Euclidean space (Liemant [1]) it seemed to be obvious that the existence of equilibria implies criticality of branching. This prejudice was disproved by the example [2] of a subcritical homogeneous branching equilibrium in dimension one. We prove that supercritical homogeneous branching processes in Euclidean space and, more general, in a broad class of topological groups have no (non – void, homogeneous) equilibria.

On completions of topological groups with respect to the maximal uniform structure and factorization of uniform homomorphisms with respect to uniform weight and dimension

A necessary and sufficient condition for completibility of topological groups with respect to the maximal uniform structure and the class of topological groups with the above-mentioned property are found. Results similar to factorization for continuous homomorphisms of R-factorizable uniform groups are extended to uniform groups.

Topological groups close to being σ-compact

Two new classes of topological groups are introduced: o-bounded and strictly o-bounded groups. We give an example of a topological group which is strictly o-bounded but is not a subgroup of a σ-compact group. A characterization of o-bounded groups in terms of their second countable continuous homomorphic images is presented. Also, we construct a topological group which iso-bounded but not strictly o-bounded.

Universal Abelian topological groups

A topological group is said to be universal in a class of topological groups if and if for every group there is a subgroup of that is isomorphic to as a topological group. A group is constructed that is universal in the class of separable metrizable topological Abelian groups.

Tychonoff property for linear groups

A criterion for a wide class of topological groups which includes linear discrete groups and Lie groups to be Tychonoff groups is established. The main result provides a criterion for an almost polycyclic group to have the Tychonoff property. By the well-known Tits alternative, this yields the required criterion for linear discrete groups. In conclusion it is pointed out that a particular case of the presented proof yields a Tychonoff property criterion for Lie groups. In addition, an example of a polycyclic group without Tychonoff subgroups of finite index is constructed.

On one-parameter subgroups in a class of topological groups

We study problems on one-parameter subgroups of infinite-dimensional groups with regular left multiplication. This class of groups contains the generalizations of Lie groups to the infinite-dimensional case, but it also contains groups which are quite different from Lie groups, for instance, large groups of homeomorphisms of manifolds. We show that the structure of one-parameter subgroups has both similarities to and differences from that of Lie groups.

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G 0, G 1 such that G 0 × G 1 is not countably compact. In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenskiĭ, is this: In ZFC there are pseudocompact, homogeneous spaces X 0, X 1 such that X 0 × X 1 is not pseudocompact; if in addition MA is assumed, the spaces X i may be chosen countably compact. Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.

On a class of topological groups more general than SIN groups

On a class of topological groups more general than SIN groups

On a class of topological groups

On a class of topological groups

Interpolation in Logiken monotoner systeme

A monotone structure ( $$\mathfrak{A} ;$$ ;μ) consists of a structure $$\mathfrak{A}$$ and a monotone systemμ over the domain of $$\mathfrak{A}$$ .L(Q n ) is $$L_{\omega \omega }$$ , enlarged by a newn-ary quantifierQ n . $$Q^n \overline x \varphi \left( {\overline x } \right)$$ says in ( $$\mathfrak{A}$$ ;μ) that there isU ∈μ such thatϕ[ā] is valid in ( $$\mathfrak{A}$$ ;μ) for allā ∈U n . If is a class of monotone structures, means thatϕ is valid in all expansions of monotone structures in . We show for the class $$\mathfrak{u}$$ of all ultrafilters that interpolation with respect to holds forL(Q n ) exactly in casen=1. Then we prove for a large class of (e.g. the class of topological groups) thatL(Q n ) satisfies interpolation with respect to for alln ≧ 1. Counterexamples indicate that the class of is sharp in some sense. Finally the results are carried over to certain topological structures and the interior quantifiersI n instead ofQ n , thereby generalizing results of Makowsky/Ziegler and Sgro, and to a multidimensional type of monotone structures including uniform spaces.

Generating varieties of topological groups

Recently several papers on varieties of topological groups have appeared. In this note we investigate the question: if Ω is a class of topological groups, what topological groups are in the variety V(Ω) generated by Ω that is, what topological groups can be “manufactured” from Ω using repeatedly the operations of taking subgroups, quotient groups and arbitrary cartesian products? We seeka general theorem which will be useful for investigating V(Ω) for well-known classesΩ.

On the Theorem of Maschke

It will be shown that the Theorem of Maschke can be carried over to certain classes of topological groups in the following way. Let A be a locally compact abelian strictly Π-divisible group, and let G be a compact totally disconnected Π-group of automorphisms of A, where Π is a set of prime numbers. If a G-admissible subgroup B of A is isolated as a direct summand, then it has a G-admissible complement in A.

Certain classes of topological groups

Certain classes of topological groups

Concepts of differentiability and analyticity on certain classes of topological groups

AbstractWe introduce the concepts of a local seminorm on a topological group and of a locally convex group, showing that discrete groups, locally compact Abelian groups and compact groups are locally convex, and that a topological vector space is locally convex as a linear space if and only if it is locally convex as a group. We show that notions of differentiability, analyticity and derivability can be defined for locally convex groups and that these notions are suitably related and well behaved. We prove that for a locally compact Abelian group G the Fourier transforms of measures of compact support on the character group Ĝ are analytic, and for G compact the coefficients of continuous irreducible unitary representations are. Using these spaces of analytic functions we define the basic concepts of a differential geometry.