Showing papers on "Topological group published in 2000"

Sofic groups and dynamical systems

Abstract: SUMMARY Sofic groups were first defined by M Gromov as a common generalization of amenable groups and residually finite groups We discuss this new class and especially its relationship to an old problem in topological dynamics of W Gottschalk on surjunctive groups The basic objects of study in topological dynamics are a pair (X;G) with X a topological space and G a group, together with a homomorphism from G into the group of homeomorphisms of X In classical dynamics G = R and the action of R is defined by the solutions of some system of dierential equations Our interest here will be in countable groups G, for which there is a very natural action defined on the compact space f1;2;¢¢¢ag G = › called the shift ae given by:

On universal minimal compact g-spaces

Abstract: For every topological group G one can define the universal minimal compact G-space X = MG characterized by the following properties: (1) X has no proper closed G-invariant subsets; (2) for every compact G-space Y there exists a G-map X → Y. If G is the group of all orientation-preserving homeomorphisms of the circle S 1 , then MG can be identified with S 1 (V. Pestov). We show that the circle cannot be replaced by the Hilbert cube or a compact manifold of dimension > 1. This answers a question of V. Pestov. Moreover, we prove that for every topological group G the action of G on MG is not 3-transitive.

Some properties of o-bounded and strictly o-bounded groups

Abstract: We continue the study of (strictly) o-bounded topological groups initiated by the first listed author and solve two problems posed earlier It is shown here that the product of a Comfort-like topological group by a (strictly) o-bounded group is (strictly) o-bounded Some non-trivial examples of strictly o-bounded free topological groups are given We also show that o-boundedness is not productive, and strict o-boundedness cannot be characterized by means of second countable continuous homomorphic images

Sequential completeness of quotient groups

Abstract: We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

Reflexively but not unitarily representable topological groups

Abstract: We show that there exists a topological group G (namely, G := L4(0;1)) such that for a certain re∞exive Banach space X the group G can be represented as a topological subgroup of Is(X) (the group of all linear isome- tries endowed with the strong operator topology) and such an X never may be Hilbert. This answers a question of V. Pestov and disproves a conjecture of A. Shtern. Let X be a real Banach space. Denote by Is(X)s (Is(X)w) the group of all linear isometries of X endowed with the strong (resp., weak) operator topologies. A representation of a Hausdorfi topological group G in X is a continuous group homomorphism G ! Is(X)s. Let K be a subclass of the class Ban of all Banach spaces. We say that G is K-representable if for a certain X 2 K there exists a topological group embedding G ,! Is(X)s. Denote by KR the class of all K- representable groups. For instance, this leads to the deflnitions of the following classes BanR, RefR, and HilbR, where Ref and Hilb denote all re∞exive and all Hilbert spaces, respectively. We say that G is re∞exively representable (unitarily representable) if G 2 RefR (resp., G 2 HilbR). Denote by TopGr the class of all Hausdorfi topological groups. We have

A note on the precompactness of weakly almost periodic groups

Abstract: MICHAEL G MEGRELISHVILI, VLADIMIR G PESTOV, AND VLADIMIR V USPENSKIJAbstract An action of a group G on a compact space X is called weakly almostperiodic if the orbit of every continuous function on X is weakly relatively compactin C(X) We observe that for a topological group G the following are equivalent: (i)every continuous action of G on a compact space is weakly almost periodic; (ii) Gis precompact For monothetic groups the result was previously obtained by Akinand Glasner, while for locally compact groups it has been known for a long time

Orbit Equivalence and Topological Conjugacy of Affine Actionson Compact Abelian Groups

Abstract: We study topological rigidity of affine actions on compact connected metrizable abelian groups. We also classify one parameter flows of translations up to orbit equivalence and discrete group actions by translations up to topological conjugacy.

Locally minimal topological groups and their embeddings into products of $o$-bounded groups

Abstract: It is proven that an infinite-dimensional Banach space (considered as an Abelian topological group) is not topologically isomorphic to a subgroup of a product of σ-compact (or more generally, o-bounded) topological groups. This answers a question of M. Tkachenko.

Moscow spaces, Pestov-Tkačenko Problem, and $C$-embeddings

Abstract: We show that there exists an Abelian topological group G such that the operations in G cannot be extended to the Dieudonne completion μG of the space G in such a way that G becomes a topological subgroup of the topological group μG. This provides a complete answer to a question of V.G. Pestov and M.G. Tkacenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the equation υX × υY = υ(X × Y ). The key role in the approach belongs to the notion of Moscow space which turns out to be very useful in the theory of C-embeddings and interacts especially well with homogeneity.

Locally free actions on Lorentz manifolds

Abstract: If a topological group G acts on a topological space M, then we say that the action is orbit nonproper provided that, for some \( m_0 \in M \),the orbit map \( g \mapsto gm_0 : G \rightarrow M \) is nonproper. In this paper we characterize the connected, simply connected Lie groups that admit a locally free, orbit nonproper action by isometries of a connected Lorentz manifold. We also consider a number of variants on this question.

Dense subsets of maximally almost periodic groups

Abstract: A (discrete) group G is said to be maximally almost periodic if the points of G are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology T on a group G is totally bounded if whenever ∅ 6= U ∈ T there is F ∈ [G] |G|, (b) each D ∈ D is dense in G, and (c) distinct D,E ∈ D satisfy |D ∩ E| < d(G); a totally bounded topological group with such a family is a strongly extraresolvable topological group. We give two theorems, the second generalizing the first. Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup. Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable. Theorem 2. Let G be maximally almost periodic. Then there are a subgroup H of G and a family D ⊆ P(H) such that (i) H is dense in every totally bounded group topology on G; (ii) the family D is a strongly extraresolvable family for every totally bounded group topology T on H such that d(H, T ) = |H|; and (iii) H admits a totally bounded group topology T as in (ii). Remark. In certain cases, for example when G is Abelian, one must in Theorem 2 choose H = G. In certain other cases, for example when the largest totally bounded group topology on G is compact, the choice H = G is impossible.

Expansiveness of algebraic actions on connected groups

Abstract: We consider endomorphism actions of arbitrary discrete semigroups on a connected metrizable topological group G. We give necessary and sufficient conditions for expansiveness of such actions when G is a Lie group or a compact finite-dimensional group.

On topological and algebraic structure of extremally disconnected semitopological groups

Abstract: Starting with a very simple proof of Frol'\i k's theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order $2$. We also apply Frol'\i k's theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every Lindelof extremally disconnected semitopological group with continuous inverse and with square roots is countable, and every extremally disconnected topological field is discrete.

Topological groups for topologists: part II

Abstract: We study the Lie algebra £°(P¿) associated with the descending central series of the puré braid group P¿, and prove that is a twisted extensión of free Lie algebras. The twisting is given by the infinitesimal braid relations.

Properties of fuzzy topological groups and semigroups

Abstract: . We characterize some basic properties of fuzzy topolog-ical groups and semigroups and show that under some conditionsin a fuzzy topological group G, x ∈A iff x ∈ ∩AU for any fuzzysubset A of G and the system {U} of all fuzzy open neighborhoodsof the identity e such that U(e) = 1. 1. Fuzzy Topological Spaces, Fuzzy Groups, and FuzzySemigroupsDefinition 1.1. A function B from a set X to the closed unitinterval [0, 1] in R is called a fuzzy set in X. For every x∈ X, B(x) iscalled a membership grade of xin B. The set {x∈ X : B(x) >0} iscalled the support of Band is denoted by supp(B).Definition 1.2. A fuzzy topology is a family T of fuzzy sets in Xwhich satisfies the following conditions:(1) ∅,X∈ T ,(2) If A,B∈ T ,then A∩ B∈ T ,(3) If A i ∈ T for each i∈ I, then ∪ i∈I A i ∈ T .T is called a fuzzy topology for X, and the pair (X,T ) is called a fuzzytopological space and is denoted by FTS for short. Every member of Tis called T -open fuzzy set. A fuzzy set is T -closed iff its complementis T -open.

Quasi-actions on trees I. Bounded valence

Abstract: Given a bounded valence, bushy tree T, we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T'. This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan's Theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first examples of two finitely generated groups which are quasi-isometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries.

Locally compact topological groups and cofinal completeness

Abstract: It is proved that a Tychonoff topological group is locally compact if and only if it is of pointwise countable type and its left uniformity is cofinally complete. From this result a characterization is derived of those T 0 paratopological groups ( X , τ) of pointwise countable type for which ( X , τ ∨ τ −1 ) is locally compact and also a characterization is deduced of locally pseudocompact topological groups in terms of cofinal completeness. Also characterized are the Tychonoff topological groups of pointwise countable type for which their left uniformity has property U . Finally, cofinal completeness of the Hausdorff–Bourbaki uniformity of a topological group is studied.

Subgroups of monothetic groups

Abstract: It is shown that every separable abelian topological group is isomorphic with a topological subgroup of a monothetic group (that is, a topological group with a single topological generator). In particular, every separable metrizable abelian group embeds into a metrizable monothetic group. More generally, we describe all topological groups that can be embedded into monothetic groups: they are exactly abelian topological groups of weight $\leq\frak c$ covered by countably many translations of every nonempty open subset.

Homogeneity of K(Ｑ)

Abstract: We prove that $\mathscr{K}(Q)$ is a topological group and characterize $\mathscr{K}(Q)$ as a first-category, zero-dimensional, separable, metrizable space of which every non-empty clopen subset is $\Pi_{1}^{1}$-complete. In particular we answer a question of Fujita and Taniyama ([5]). With the additional assumption of Analytic Determinacy it was proved in [5] that $\mathscr{K}(Q)$ is a homogeneous space.

Residual finiteness of groups and topological dynamics

Abstract: The property of residual finiteness of finitely generated groups admits an interpretation in terms of topological dynamics. In this paper it is proved that a finitely generated group is residually finite if and only if it is isomorphic to a group whose action on some compact set is efficient and distal.

Measurability of Automorphisms of Topological Groups

Abstract: Relations between the measurability and continuity of algebraic automorphisms of topological groups depending on the types of groups are examined. Various cases are considered and theorems on the continuity of measurable automorphisms are proved; for instance, such theorems are proved for separable locally compact groups and automorphisms measurable with respect to nonnegative Haar measures. On the other hand, examples of nonmetrizable nonseparable compact groups with Haar measures and of non-locally-compact separable metrizable groups with measures μ quasi-invariant with respect to dense subgroups admittings μ-measurable discontinous automorphisms are given.