Self-duality in the class of precompact groups

A topological Abelian group G is called (strongly) self-dual if there exists a topological isomorphism Φ:G→G^∧ of G onto the dual group G^∧ (such that Φ(x)(y)=Φ(y)(x) for all x,y∈G). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κω=κ, there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.     

The arithmetical hierarchy of nilpotent torsion-free groups

Previously, N. Khisamiev proved that all {ie172-1} Abelian torsion-free groups are {ie172-2}. We prove that for the class of nilpotent torsion-free groups, the situation is different: even the quotient group F of a {ie172-3} nilpotent group of class 2 by its periodic part may fail to have a {ie172-4}. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 308–313, May–June, 1996.     

A general duality result for precompact sets

In this paper we present an elementary proof of a general duality result for precompact sets which can be considered as a far-reaching generalization of a well-known result of Grothendieck on precompactness in dual systems. It is then shown that a number of known results can be deduced from it, amongst others a general form of the Arzela-Ascoli theorem and Grothendieck's duality theorem itself.     

Homomorphism reductions on Polish groups

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if G is a Polish group and \(H,L \subseteq G\) are subgroups, we say H is homomorphism reducible to L iff there is a continuous group homomorphism \(\varphi : G \rightarrow G\) such that \(H = \varphi ^{-1} (L)\). We previously showed that there is a \(K_\sigma \) subgroup L of the countable power of any locally compact Polish group G such that every \(K_\sigma \) subgroup of \(G^\omega \) is homomorphism reducible to L. In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval.     

Polish topologies for graph products of cyclic groups

We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from [8], [9] and [4].     

Borel subgroups of Polish groups

We study three classes of subgroups of Polish groups: Borel subgroups, Polishable subgroups, and maximal divisible subgroups. The membership of a subgroup in each of these classes allows one to assign to it a rank, that is, a countable ordinal, measuring in a natural way complexity of the subgroup. We prove theorems comparing these three ranks and construct subgroups with prescribed ranks. In particular, answering a question of Mauldin, we establish the existence of Borel subgroups which are -complete, α?3, and -complete, α?2, in each uncountable Polish group. Also, for every α<ω₁ we construct an Abelian, locally compact, second countable group which is densely divisible and of Ulm length α+1. All previously known such groups had Ulm length 0 or 1.     

On surjectively universal Polish groups

A Polish group is surjectively universal if it can be continuously homomorphically mapped onto every Polish group. Making use of a type of new metrics on free groups by Ding and Gao (2007) [3], we prove the existence of surjectively universal Polish groups, answering in the positive a question of Kechris. In fact, we give several examples of surjectively universal Polish groups.We find a sufficient condition to guarantee that the new metrics on free groups can be computed directly. We also compare this condition with CLI groups.     

Pairings and actions for dynamical quantum groups

Dynamical quantum groups constructed from a FRST-construction using a solution of the quantum dynamical Yang-Baxter equation are equipped with a natural pairing. The interplay of the pairing with *-structures, corepresentations and dynamical representations is studied, and natural left and right actions are introduced. Explicit details for the elliptic U(2) dynamical quantum group are given, and the pairing is calculated explicitly in terms of elliptic hypergeometric functions. Dynamical analogues of spherical and singular vectors for corepresentations are introduced.     

Characterizing sequences for precompact group topologies

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact  ) if there is a sequence u=(_un)$\mathbf{u}=(u_n)$ in G such that τ is the finest precompact group topology on G   making u=(_un)$\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group (G,τ)$(G,\tau )$ is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ   and the groups (G,τ)$(G,\tau )$ and (G,η)$(G,\eta )$ have the same Pontryagin dual groups (in other words, (G,τ)$(G,\tau )$ is not a Mackey group in the class of maximally almost periodic groups).     

Vaught's conjecture and the Glimm-Effros property for Polish transformation groups

We extend the original Glimm-Effros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture.

     

Non-nesting actions of Polish groups on real trees

We prove that if a Polish group G with a comeagre conjugacy class has a non-nesting action on an R-tree, then every element of G fixes a point.     

Homogeneous spaces and transitive actions by Polish groups

We prove that for every homogeneous and strongly locally homogeneous Polish space X there is a Polish group admitting a transitive action on X. We also construct an example of a homogeneous Polish space which is not a coset space and on which no separable metrizable topological group acts transitively.     

Borel globalizations of partial actions of Polish groups

We show that the enveloping space \({\mathbb {X}}_G\) of a partial action of a Polish group G on a Polish space \({\mathbb {X}}\) is a standard Borel space, that is to say, there is a topology \(\tau \) on \({\mathbb {X}}_G\) such that \(({\mathbb {X}}_G, \tau )\) is Polish and the quotient Borel structure on \({\mathbb {X}}_G\) is equal to \(Borel({\mathbb {X}}_G,\tau )\). To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught’s transform are valid for partial actions of groups.     

On the hierarchy of cohomological dimensions of groups

Using Auslander’s G-dimension, we assign a numerical invariant to any group Γ. It provides a refinement of the cohomological dimension and fits well into the well-known hierarchy of dimensions assigned already to Γ. We study this dimension and show its power in reflecting the properties of the underlying group. We also discuss its connections to relative and Tate cohomology of groups.     

Generating ranking groups in the analytical hierarchy process

The paper introduces a convenient procedure of ranking N alternatives through direct comparisons in AHP. The alternatives are divided into groups in such a way that dominant relationship exists between the groups but not among the alternatives within each group. This method is suitable for situations where the strict ranking in a sequence for all alternatives is not reliable or not necessary. Two procedures are proposed to construct the AHP ranking groups. The proposed grouping procedures can be used in conjunction with the traditional approaches.     

Generic elements in isometry groups of Polish ultrametric spaces

This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) has a neighborhood basis at the identity consisting of open subgroups with ample generics. It also gives a characterization of the existence of an open subgroup in Iso(X) with a comeager conjugacy class.We also study the transfinite sequence defined by the projection of a Polish ultrametric space X on the ultrametric space of orbits of X under the action of Iso(X).     

Strong measure zero in separable metric spaces and Polish groups

The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer–Specker group \({{\mathbb{Z}^{\omega}}}\). The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and Steprāns (Ann Pure Appl Logic 140(1–3):52–59, 2006).