Locally compact abelian groups with symplectic self-duality

Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer).     

Locally compact groups with complementary closed subgroups

Topological groups with complementary closed abelian subgroups and with complementary monothetic subgroups are considered.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 641–649, December, 1969.     

Locally compact unimodular groups with atomic duals

Milan Journal of Mathematics - A locally compact commutative group is compact if and only if its dual group is discrete, in other words if its regular representation is atomic. Nevertheless, there...     

Locally compact groups which are separably categorical structures

We describe locally compact groups which are separably categorical metric structures.     

Locally normal subgroups of simple locally compact groups

We announce various results concerning the structure of compactly generated simple locally compact groups. We introduce a local invariant, called the structure lattice, which consists of commensurability classes of compact subgroups with open normaliser, and show that its properties reflect the global structure of the ambient group.     

Locally compact abelian groups with trivial multiplier group

LCA groups with trivial multiplier group are characterized. It is found that every LCA group with nontrivial multiplier group has an “irreducible” quotient with the same property, and the irreducible groups are described.     

Locally compact groups acting on trees and propertyT

FollowingKazhdan, a separable locally compact groupG is said to have propertyT if the trivial representation is isolated in the dual space,, of equivalence classes of continuous irreducible unitary representations ofG. We generalize results ofMargulis—Tits by showing that groups which have propertyT can not be amalgams.Research supported by NSF.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Locally compact connected groups acting on euclidean space with lie isotropy groups are lie

It is shown that a locally compact connected group must be a Lie group if it acts on a space with certain local and global cohomological properties (for example, euclidean space and even-dimensional spheres) in such a way that all isotropy groups are Lie groups. Apart from the obvious application to free actions, this result can also be applied to automorphism groups of continous incidence geometries of a particular kind (four-dimensional stable planes).     

Locally compact (2, 2)‐transformation groups

We determine all locally compact imprimitive transformation groups acting sharply 2‐transitively on a non‐totally disconnected quotient space of blocks inducing on any block a sharply 2‐transitive group and satisfying the following condition: if Δ₁, Δ₂ are two distinct blocks and P_i, Q_i ∈ Δ_i (i = 1, 2), then there is just one element in the inertia subgroup which maps P_i onto Q_i. These groups are natural generalizations of the group of affine mappings of the line over the algebra of dual numbers over the field of real or complex numbers or over the skew‐field of quaternions. For imprimitive locally compact groups, our results correspond to the classical results of Kalscheuer for primitive locally compact groups (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Locally compact multivector extensions

The aim of this paper is to develop a locally compact extension of an arbitrary normed space in such a way that the initial algebraic structure is prolonged in some sense. To obtain such an extension, we weaken vector space axioms allowing a set-valued addition and introduce in this scheme a topological structure, by means of a hypertopology, and a compatible proximity. Finally, the locally compact multivector extension appears as an ultrafilter space. We also provide a Young measure related interpretation of these extensions when the normed space is an L^p space.