Asymptotic dimension and small subsets in locally compact topological groups (original) (raw)

Some properties of locally compact groups

Journal of the Australian Mathematical Society, 1967

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.

Topologies on locally compact groups

Bulletin of The Australian Mathematical Society, 1988

Using the Iwasawa structure theorem for connected locally compact Hausdorff groups we show that every locally compact Hausdorff group G is homeomorphic to R n x K x D, where n is a non-negative integer, if is a compact group and D is a discrete group. This makes recent results on cardinal numbers associated with the topology of locally compact groups more transparent. For abelian G, we note that the dual group, G, is homeomorphic to R n x K X D. This leads us to the relationship card G = U» 0 (G) + 2"o(<3) > w u e r e u> (respectively, wo ) denotes the weight (respectively local weight) of the topological group. From this classical results such as card G -1 c a a for compact Hausdorff abelian groups, and u>(G) = <*(G\ for general locally compact Hausdorff abelian groups are easily derived.

Generators for locally compact groups

Proceedings of The Edinburgh Mathematical Society, 1993

It is proved that if G is any compact connected Hausdorff group with weight w(G)gc, R is the topological group of all real numbers and n is a positive integer, then the topological group GxR" can be topologically generated by n + 1 elements, and no fewer elements will suffice.

Density in locally compact topological groups

Topology and its Applications

We introduce some notions of density in a locally compact topological group G which extend some notions in discrete semigroups in which density is based on nets of finite sets. The new notions are based on nets of compact sets. We consider that the LUCLUC-compactification of G is a set of z-filters on LUC(G)LUC(G) and obtain new information about the algebraic structure of this form. Also we present a version of Schurʼs Theorem in a locally compact topological group.

Locally minimal topological groups 2

Journal of Mathematical Analysis and Applications, 2011

The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G, τ ) is called locally minimal if there exists a neighborhood U of 0 in τ such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than τ. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups.

A decomposition theorem for locally compact groups

Georgian Mathematical Journal

We prove that, under certain restrictions, every locally compact group equipped with a nonzero, σ-finite, regular left Haar measure can be decomposed into two small sets, one of which is small in the sense of measure and the other is small in the sense of category, and all such decompositions originate from a generalised notion of a Lebesgue point. Incidentally, such class of topological groups for which this happens turns out to be metrisable. We also observe an interesting connection between Luzin sets in such spaces and decompositions of the above type.

Convergence Groups, Hausdorff Dimension, and a Theorem of Sullivan and Tukia

Geometriae Dedicata, 2004

We show that a discrete, quasiconformal group preserving H n has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity to H n. This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on H n , i.e. is a Kleinian group. From this generalization we are able to extract geometric information about infinite-index subgroups within certain of these groups.

Locally minimal topological groups

2009

A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudo--convex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. We also prove that a bounded abelian group GGG admits a non-discrete locally minimal and locally GTG group topology iff ∣G∣geqcont|G|\geq \cont∣G∣geqcont.

A Note on Locally Compact Subsemigroups of Compact Groups

arXiv (Cornell University), 2020

An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.

Locally compact groups and locally minimal group topologies

Fundamenta Mathematicae, 2019

Minimal groups are the Hausdorff topological groups G satisfying the open mapping theorem with respect to continuous isomorphisms, i.e., every continuous isomorphism G / / H, with a Hausdorff topological group H, is a topological isomorphism. A topological group (G, τ) is called locally minimal if there exists a neighbourhood V of the identity such that for every Hausdorff group topology σ ≤ τ with V ∈ σ one has σ = τ. Minimal groups, as well as locally compact groups, are locally minimal. According to a well known theorem of Prodanov every subgroup of an infinite compact abelian group K is minimal if and only if K is isomorphic to the group Z p of p-adic integers for some prime p. We find a remarkable connection of local minimality to Lie groups and p-adic numbers by means of the following results extending Prodanov's theorem: every subgroup of a locally compact abelian group K is locally minimal if and only if K is either a Lie group or K has an open subgroup isomorphic to Z p for some prime p. In the nonabelian case we prove that all subgroups of a connected locally compact group are locally minimal if and only if K is a Lie group, resolving in the positive Problem 7.49 from [?].