Showing papers on "Topological group published in 2003"

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.

On the Universal Space for Group Actions with Compact Isotropy

Abstract: LetG be a locally compact topological group and EG its universal space for the family of compact subgroups. We give criteria for this space to be G-homotopy equivalent to a d-dimensional G-CW -complex, a finite G-CW -complex or a G-CW -complex of finite type. Essentially we reduce these questions to discrete groups, and to the homological algebra of the orbit category of discrete groups with respect to certain families of subgroups.

The universal minimal system for the group of homeomorphisms of the cantor set

Abstract: Each topological group G admits a unique universal minimal dy- namical system (M(G);G). For a locally compact non-compact group this is a nonmetrizable system with a rich structure, on which G acts eectively. However there are topological groups for which M(G) is the trivial one point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. In this paper we show that for the topological group G = Homeo(E) of self home- omorphisms of the Cantor set E, with the topology of uniform convergence, the universal minimal system (M(G);G) is isomorphic to Uspenskij's 'max- imal chains' dynamical system (';G) in 22 E . In particular it follows that M(G) is homeomorphic to the Cantor set. Our main tool is the 'dual Ram- sey theorem', a corollary of Graham and Rothschild's Ramsey's theorem for n-parameter sets. This theorem is used to show that every minimal symbolic G-system is a factor of (';G) and then a general procedure for analyzing G- actions of zero dimensional topological groups is used to show that (M(G);G) is isomorphic to (';G).

A Unified Approach To The Concept Of Fuzzy L-Uniform Space

Abstract: The theory of uniform structures is an important area of topology which in a certain sense can be viewed as a bridge linking metrics as well as topological groups with general topological structures. In particular, uniformities form, the widest natural context where such concepts as uniform continuity of functions, completeness and precompactness can be extended from the metric case. Therefore, it is not surprising that the attention of mathematicians interested in fuzzy topology constantly addressed the problem to give an appropriate definition of a uniformity in fuzzy context and to develop the corresponding theory. Already by the late 1970’s and early 1080’s, this problem was studied (independently at the first stage) by three authors: B. Hutton [21], U. Hohle [11, 12], and R. Lowen [30]. Each of these authors used in the fuzzy context a different aspect of the filter theory of traditional uniformities as a starting point, related in part to the different approaches to traditional unformities as seen in [37, 2] vis-a-vis [36, 22]; and consequently, the applied techniques and the obtained results of these authors are essentially different. Therefore it seems natural and urgent to find a common context as broad as necessary for these theories and to develop a general approach containing the previously obtained results as special cases—it was probably S. E. Rodabaugh [31] who first stated this problem explicitly.

The Continuous Wavelet Transform and Symmetric Spaces

Abstract: The continuous wavelet transform has become a widely used tool in applied science during the last decade. In this article we discuss some generalizations coming from actions of closed subgroups of GL(n, R) acting on R n . In particular, we propose a way to invert the wavelet transform in the case where the stabilizer of a generic point in R n is not compact, but a symmetric subgroup, a case that has not previously been discussed in the literature. Introduction, Wavelets on ax +b-group The continuous wavelet transform has become a widely used tool in applied science during the last decade. The best known example deals with wavelets on the real line. Each wavelet by taking "matrix coefficients", i.e., inner products with translations and dilations of the wavelet, defines a wavelet transform which can be used to reconstruct the function from the dilations and translations of the wavelet. This process helps compensate for the local-nonlocal behavior of the Fourier transform. Translations and dilations form the so-called ax + b-group of transformations of the real line. These act in a natural manner as unitary operators on L 2 (R). In that way, wavelet transforms are simply a part of the representation theory of the ax + b-group. This observation is the basis for the generalization of the continuous wavelet transform to higher dimensions and more general settings. There have been further attempts to generalize these ideas to arbitrary groups; see (4) and the references therein. In this article we review some of the basic ideas for the general wavelet transform for groups acting on R n . We start by reviewing the classical one dimensional wavelet transform. In this simple setting, the usual definition of a wavelet is equivalent to the corresponding matrix coefficients being square integrable on theax+b-group. A natural generalization of this to an arbitrary representation (�, H) of a topological group G is to say, that a vector u ∈ H \ {0} is a wavelet if a 7→ (v | �(a)u) is square integrable for all v ∈ H. The generalized wavelet transform is then Wv(a) := (v | �(a)v). The basic facts for this transform and, in particular, the inversion formula are presented in section 1. In section 2 these notions are applied to topological groups H acting on R n by a representation �. Let G = H ×� R n be the semi-direct product of H and R n . Then G acts on R n and L 2 (R n ). When the action of H on R n is sufficiently well behaved, the decomposition of L 2 (R n ) under G can be described in terms of the orbits in R n under the transpose (contragredient) representation � ' where � ' (a) = �(a −1 ) t . In general, one

Transfer methods for o-minimal topology

Abstract: Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties which are transferred from φ M to φ R and vice versa. Then, we apply these transfer results to give a new proof of a result of M. Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.

A property of Dunford-Pettis type in topological groups

Abstract: The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory. In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups. For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.

On a universality property of some abelian Polish groups

Abstract: We show that every abelian Polish group is the topological factor group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the unitary group.

Noncommutative localization in topology

Abstract: A survey of the applications of the noncommutative Cohn localization of rings to the topology of manifolds with infinite fundamental group, with particular emphasis on the algebraic K- and L-theory of generalized free products.

Chaotic group actions

Abstract: It is proved that there is no chaotic group actions on any topological space with free arc. In this paper the chaotic actions of the group like G × F, where F is a finite group, are studied. In particular, under a suitable assumption, if F is a cyclic group, then the topological space which admits a chaotic action of Z × F must admit a chaotic homeomorphism. A topological space which admits a chaotic group action but admits no chaotic homeomorphism is constructed.

Ubiquity of Free Subgroups

Abstract: Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.

FC^--elements in totally disconnected groups and automorphisms of infinite graphs

Abstract: An element in a topological group is called an $\mathrm{FC}^-$-element if its conjugacy class has compact closure. The $\mathrm{FC}^-$-elements form a normal subgroup. In this note it is shown that in a compactly generated totally disconnected locally compact group this normal subgroup is closed. This result answers a question of Ghahramani, Runde and Willis. The proof uses a result of Trofimov about automorphism groups of graphs and a graph theoretical interpretation of the condition that the group is compactly generated.

A mackey-arens theorem for topological abelian groups

Abstract: If G is an Abelian group with a collection X of homomorphisms into the usual torus T such that (a) X is a group urider pointwise operation, and (b) the weakest topology on G that maltes the elements of X continuous is Hausdorff. We give necessary and sufficient conditions for the existence of the finest locally quasi-convex group topology on G, having exactly X as the group of continuous homomorphisms into T. Such topology is the topoiogy of uniform convergence on an appropriate, sometimes proper, subfamily of weakly compact quasi-convex subsets of X. These results are motivated by the usual Mackey-Arens theorem for locally convex spaces. 1. Introduction and motivation Given an Abelian topological group (G, t), with underlying group G and topology t, we say that a group topology A on G is compatible with t if the sets of t- and A-continuous homomorphisms from G into T coincide, Le., if (G, tT — (G, A)". For example, it is a theorem of Comfort and Ross (8) that cr, the weakest topology on G that makes the elements of (G, tT continuous, is compatible with (G, t). Following the lead from the theory of locally convex spaces (LCSs), it is natural to ask (a) if (G, i) has its Mackey topology, i.e., the finest group topology ?i on G compatible with t, in the sense that if A is a com- patible group topology with t, then A - they give conditions for its existence, and show that for certain types of groups the Mackey topology exists. As for question (b) the contributions of (5) are signifi- cant as they look at the family € of weakly compact quasiconvex subsets of the character group -the exact counterpart in the theory of LCS- giving necessary and sufficient conditions for £ to be the collection requested in (b). In this paper we give a positive answer to question (b), showing that if the (locally quasi-convex) Mackey topology \L exists, then it is given as the topology

Projective Limits of Finite-Dimensional Lie Groups

Abstract: For a topological group $G$ we define $\cal N$ to be the set of all normal subgroups modulo which $G$ is a finite-dimensional Lie group. Call $G$ a pro-Lie group if, firstly, $G$ is complete, secondly, $\cal N$ is a filter basis, and thirdly, every identity neighborhood of $G$ contains some member of $\cal N$. It is easy to see that every pro-Lie group $G$ is a projective limit of the projective system of all quotients of $G$ modulo subgroups from $\cal N$. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup $N$ of a pro-Lie group $G$, for any one parameter subgroup $Y \colon \mathbb{R} \to G/N$ there is a one parameter subgroup $X \colon \mathbb{R}\to G$ such that $X(t) N = Y(t)$ for any real number $t$. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.

On subgroups of saturated or totally bounded paratopological groups

Abstract: A paratopological group G is saturated if the in- verse U−1 of each non-empty set U ½ G has non-empty interior. It is shown that a [first-countable] paratopological group H is a closed subgroup of a saturated (totally bounded) [abelian] paratopological group if and only if H admits a continuous bijective homomorphism onto a (totally bounded) [abelian] topological group G [such that for each neighborhood U ½ H of the unit e there is a closed subset F ½ G with e 2 h−1(F) ½ U]. As an application we construct a paratopological group whose character exceeds its ¼-weight as well as the character of its group reflexion. Also we present several ex- amples of (para)topological groups which are subgroups of totally bounded paratopological groups but fail to be subgroups of regular totally bounded paratopological groups.

Two countably compact topological groups: One of size ℵ_{} and the other of weight ℵ_{} without non-trivial convergent sequences

Abstract: E. K. van Douwen asked in 1980 whether the cardinality of a countably compact group must have uncountable cofinality in ZFC. He had shown that this was true under GCH. We answer his question in the negative. V. I. Malykhin and L. B. Shapiro showed in 1985 that under GCH the weight of a pseudocompact group without non-trivial convergent sequences cannot have countable cofinality and showed that there is a forcing model in which there exists a pseudocompact group without non-trivial convergent sequences whose weight is w 1 < c. We show that it is consistent that there exists a countably compact group without non-trivial convergent sequences whose weight is N w .

An Abstract Approach to Sampling-Type Operators inspired by the Work of P.L.Butzer

Abstract: Here a new approach to the study of approximation problems by means of a family of linear integral operators is presented. The abstract approach used, which is based mainly on the use of locally compact topological groups, allows us to give a unifying method in order to study a large class of approximation problems and, as an application to signal processing, in the setting of modular spaces we obtain the reconstruction of signals which can be discontinuous and are not necessarily of finite energy or band-limited.

A Compact Homogeneous S-space

Abstract: Under the continuum hypothesis, there is a compact homogeneousstrong S-space. 1 Introduction A space X is hereditarily separable ( HS ) iff every subspace is separable. An S -space is a regular Hausdorff HS space with a non-Lindel¨of subspace. A spaceX is homogeneous iff for every x,y ∈ X there is a homeomorphism f of Xonto X with f(x) = y. Under CH , several examples of S-spaces have beenconstructed, including topological groups (see [4]) and compact S-spaces (see[7]). It is asked in [1, 5] whether there are compact homogeneous S-spaces. Aswe shall show in Theorem 4.2, there are under CH . This cannot be done in ZFC , since there are no compact S-spaces under MA + ¬ CH (see [12]); thereare no S-spaces at all under PFA (see [13]).In Section 2, we use a slightly modified version of the construction in [7, 10]to refine the topology of any given second countable space, and turn it into afirst countable strong S -space (i.e., each of its finite powers is an S-space). InSection 3, we show that if the original space is compact, then there is a naturalcompactification of the new space which is also a first countable strong S-space.If in addition the original space is zero-dimensional, then the ω

Cumulants are universal homomorphisms into Hausdorff groups

Abstract: This is a contribution to the theory of sums of independent random variables at an algebraico-analytical level: Let ProbOpen image in new window denote the convolution semigroup of all probability measures on Open image in new window with all moments finite, topologized by polynomially weighted total variation. We prove that the cumulant sequence Open image in new window regarded as a function from ProbOpen image in new window into the additive topological group Open image in new window ofall real sequences, is universal among continuous homomorphisms from ProbOpen image in new window into Hausdorff topological groups, in the usual sense that every other such homomorphism factorizes uniquely through κ. An analogous result, referring to just the first Open image in new window cumulants,holds for the semigroup Open image in new window of all probability measures with existing rth moments. In particular, there is no nontrivial continuous homomorphism from Open image in new window the convolution semigroup of all probability measures, topologized by the total variation metric, into any Hausdorff topological group.

Dynamical Systems, Topology and Conductivity in Normal Metals in strong magnetic fields

Abstract: We represent here the full description of all asymptotic regimes of conductivity behavior in the so-called "Geometric Strong Magnetic Field limit" in the 3D single crystal normal metals with topologically complicated Fermi surfaces. In particular, new observable integer-valued characteristics of conductivity of the topological origin were introduced by the present authors few years ago; they are based on the Topological Resonance found by the present authors and play the basic role in the total picture. Our investigation is based on the study of dynamical systems on Fermi surfaces for the semi-classical motion of electron in magnetic field realized by the Moscow topological group.

Banach-Dieudonné Theorem Revisited

Abstract: We prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonne, which is well known in the theory of locally convex spaces. We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have called the quasi-convex compactness property, or briefly qcp (Section 3).

An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup

Abstract: Conjecturally, for p an odd prime and R a certain ring of p-integers, the stable general linear group GL(R) and the etale model for its classifying space have isomorphic mod p cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if p is regular and certain homology classes for SL 2 (R) vanish. We check that this criterion is satisfied for p = 3 as evidence for the conjecture.

Local compactness in free topological groups

Abstract: We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω if and only if A2(X) is locally compact if an only if F2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F(X) is locally compact for each n ∈ ω if and only if F4(X) is locally compact if and only if Fn(X) has pointwise countable type for each n ∈ ω if and only if F4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω if and only if A2(X) has pointwise countable type if and only if F2(X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that F2(X) is locally compact if and only if F3(X) is locally compact, and that F2(X) has pointwise countable type if and only if F3(X) has pointwise countable type.

$ K_2$ for the simplest integral group rings and topological applications

Abstract: We calculate the group , where is the group ring of a fundamental group with coefficients in the field and is the simplest elementary Abelian group of rank 2. Using these calculations we estimate from below the value , where is the integral group ring of the group . This calculation yields certain corollaries in the theory of pseudo-isotopies, since the group turns out to be non-trivial. Constructions in differential topology are discussed that lead to calculations of -valued invariants.

The mapping class group of a disk with infinitely many holes

Abstract: A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.

Exercises in Abelian Group Theory

Abstract: Preface. List of Symbols. I: Statements. 1. Basic notions. Direct sums. 2. Divisible groups. 3. Pure subgroups. Basic subgroup. 4. Topological groups. Linear topologies. 5. Algebraically compact groups. 6. Homological methods. 7. p-groups. 8. Torsion-free groups. 9. Mixed groups. 10. Subgroup lattices of groups. II: Solutions. 1. Basic notions. Direct sums. 2. Divisible groups. 3. Pure subgroups. Basic subgroups. 4. Topological groups. Linear topologies. 5. Algebraically compact groups. 6. Homological methods. 7. p-groups. 8. Torsion-free groups. 9. Mixed groups. 10. Subgroup lattices of groups. Bibliography. Index.