Zusammenhängende Untergruppen von pro-Lieschen Gruppen

In this paper we consider the lattice G of all closed connected subgroups of pro-Lie groups G, which seems to have in some sense a more geometric nature than the full lattice of all closed subgroups. We determine those pro-Lie groups whose lattice shares one of the elementary geometric lattice properties, such as the existence of complements and relative complements, semi-modularity and its dual, the chain condition, self-duality and related ones. Apart from these results dealing with subgroup lattices we also get two structure theorems, one saying that maximal closed analytic subgroups of Lie groups actually are maximal among all analytic subgroups, the other that each connected abelian pro-Lie group is a direct product of a compact group with copies of the reals.     

Projective Limits of Finite-Dimensional Lie Groups

For a topological group G we define N to be the set of all normalsubgroups modulo which G is a finite-dimensional Lie group.Call G a pro-Lie group if, firstly, G is complete, secondly,N is a filter basis, and thirdly, every identity neighborhoodof G contains some member of N. It is easy to see that everypro-Lie group G is a projective limit of the projective systemof all quotients of G modulo subgroups from N. The converseimplication emerges as a difficult proposition, but it is shownhere that any projective limit of finite-dimensional Lie groupsis a pro-Lie group. It is also shown that a closed subgroupof a pro-Lie group is a pro-Lie group, and that for any closednormal subgroup N of a pro-Lie group G, for any one parametersubgroup Y : R G/N there is a one parameter subgroup X : R G such that X(t) N = Y(t) for any real number t. The categoryof all pro-Lie groups and continuous group homomorphisms betweenthem is closed under the formation of all limits in the categoryof topological groups and the Lie algebra functor on the categoryof pro-Lie groups preserves all limits and quotients. 2000 MathematicsSubject Classification 22E65, 22D05, 22E20, 22A05, 54B35.     

Algebraic subgroup lattices of topological groups

It is shown that the lattice of all closed subgroups of a compact topological group and the lattice of all connected closed subgroups of a pro-Lie group are algebraic, even arithmetic, if they are equipped with the order opposite to the natural one. The compact elements form ideals in these lattices and are explicitly determined. In the course of the proof the question is treated whether forming lattices of closed subgroups of topological groups commutes with projective limits.Presented by Laszlo Fuchs.     

Quasi‐convexly dense and suitable sets in the arc component of a compact group

Let G be an abelian topological group. The symbol $\widehat{G}Let G be an abelian topological group. The symbol $\widehat{G}$ denotes the group of all continuous characters $\chi :G\rightarrow {\mathbb T}$ endowed with the compact open topology. A subset E of G is said to be qc‐dense in G provided that χ(E)?φ([? 1/4, 1/4]) holds only for the trivial character $\chi \in \widehat{G}$, where $\varphi : {\mathbb R}\rightarrow {\mathbb T}={\mathbb R}/{\mathbb Z}$ is the canonical homomorphism. A super‐sequence is a non‐empty compact Hausdorff space S with at most one non‐isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super‐sequence converging to 0 that is qc‐dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism $r:\widehat{G}\rightarrow \widehat{G}_a$ defined by $r(\chi )=\chi \upharpoonright _{G_a}$ for $\chi \in \widehat{G}$, is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component G_a contains a super‐sequence converging to the identity that is qc‐dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group.     

Locally minimal topological groups 2

We continue in this paper the study of locally minimal groups started in Außenhofer et al. (2010) [4]. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group G almost minimal if it has a closed, minimal normal subgroup N such that the quotient group G/N is uniformly free from small subgroups. The class of almost minimal groups includes all locally compact groups, and is contained in the class of locally minimal groups. On the other hand, we provide examples of countable precompact metrizable locally minimal groups which are not almost minimal. Some other significant properties of this new class are obtained.     

On Baire Measurable Homomorphisms of Quotients of the Additive Group of the Reals

The quotient ℝ/G of the additive group of the reals modulo a countable subgroup G does not admit nontrivial Baire measurable automorphisms.     

Flat rank of automorphism groups of buildings

The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities alg-rk(W) ≤ flat-rk(G) ≤ rk(|W|₀). Here, alg-rk(W) is the maximal Z-rank of abelian subgroups of W, and rk(|W|₀) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|₀. We can prove these inequalities under weaker assumptions. We also show that for any integer n ≥ 1 there is a simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G) = n and which is not linear.     

Product decompositions of solvable Lie groups

Summary The article investigate the structure of real solvable connected Lie groups. It is described how one can decompose a solvable Lie group in direct and semidirect products of closed connected subgroups. In particular, the commutator group, Cartan subgroups, the center, maximal compactly embedded subgroups and tori are considered. Furthermore, one can find special solvable Lie groups and their product decompositions, namely compactly generated solvable Lie groups and those Lie groups which are generated by maximal compactly embedded subgroups. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Final group topologies,Kac-Moody groups and Pontryagin duality

We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k _ω-space, or locally k _ω. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of k _ω-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally k _ω topological abelian groups, which are dual to each other. In particular, we explore the relations between countable projective limits of almost metrizable abelian groups and countable direct limits of locally k _ω abelian groups.     

Locally minimal topological groups 1

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. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. 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 pseudoconvex 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. Unlike the minimal topologies, the locally minimal ones are always available on “large” groups. To support this line, we prove that a bounded abelian group G admits a non-discrete locally minimal and locally GTG group topology iff |G|?c.     

4-Dimensionale projektive Ebenen mit grosser abelscher Kollineationsgruppe

We show that a 4-dimensional connected abelian group can act in exactly five different ways as a collineation group of a compact 4-dimensional projective plane. Furthermore the complex projective plane is characterized as the only compact 4-dimensional projective plane which admits two different 4-dimensional abelian collineation groups.

---

---

Herrn Professor Dr. Eelmut Karzel zum 60. Geburtstag     

Essential subgroups of topological groups

We study the class Wof Hausdorff topological groups Gfor which the following two cardinal invariants coincide

ES(G)=min{|H|:H≤ Gdense and essential}

TD(G)=min{|H|:H≤ Gtotally dense}

We prove that W contains the following classes:locally compact abelian groups, compact connected groups, countably compact totally discon¬nected abelian groups, topologically simple groups, locally compact Abelian groups when endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to giv ean explicit computation of the common value of ESand TD.     

Compactly Generated Relative Stable Categories

Let G be a finite group. The stable module category of G has been applied extensively in group representation theory. In particular, it has been used to great effect that it is a triangulated category which is compactly generated by the class of finitely generated modules. Let H be a subgroup of G. It is possible to define a stable module category of G relative to H. This is also a triangulated category, but no non-trivial examples have been known where it was compactly generated. While the finitely generated modules are compact objects, they do not necessarily generate the category. We show that the relative stable category is compactly generated if the group algebra of H has finite representation type. In characteristic p, this is equivalent to the Sylow p-subgroups of H being cyclic.     

Adelic version of Margulis arithmeticity theorem

We generalize Margulis's S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one and set . Let S be any subset of R. For each , let be a connected semisimple adjoint -group and be a compact open subgroup for each finite prime . Let denote the restricted topological product of 's, with respect to 's. Note that if S is finite, . We show that if , any irreducible lattice in is a rational lattice. We also present a criterion on the collections and for to admit an irreducible lattice. In addition, we describe discrete subgroups of generated by lattices in a pair of opposite horospherical subgroups. Received: 30 November 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001     

Splitting Mixed Groups of Finite Torsion-Free Rank

Abstract

First, we give a necessary and sufficient condition for torsion-free finite rank subgroups of arbitrary abelian groups to be purifiable. An abelian group G is said to be a strongly ADE decomposable group if there exists a purifiable T(G)-high subgroup of G. We use a previous result to characterize ADE decomposable groups of finite torsion-free rank. Finally, in an extreme case of strongly ADE decomposable groups, we give a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.     

Malnormal subgroups of lattices and the Pukánszky invariant in group factors

Let G be a connected semisimple real algebraic group. Assume that G(R) has no compact factors and let Γ be a torsion-free uniform lattice subgroup of G(R). Then Γ contains a malnormal abelian subgroup A. This implies that the II₁ factor VN(Γ) contains a masa A with Pukánszky invariant {∞}.     

The finite torsion subgroup of an LCA group need not split

A locally compact abelian topological groupG is constructed whose maximal torsion subgroup is finite but not a topological direct summand ofG.     

A Note on Large Characteristic Subgroups

It is known that if a group G has an abelian subgroup of finite index n, then it contains an abelian characteristic subgroup A of index at most nⁿ. The aim of this paper is to improve this bound by showing that the characteristic subgroup A can be chosen of index at most n². Examples prove that this bound is the best possible. Our main result is obtained as an application of a general method for the construction of large characteristic subgroups.     

A torsion-free abelian group of finite rank exists whose quotient group modulo the square subgroup is not a nil-group

The first example of a finite rank torsion-free abelian group A such that the quotient group of A modulo the square subgroup of A is not a nil-group is indicated (in both cases of associative and general rings). In particular, the answer to the question posed by A.E. Stratton and M.C. Webb in [18], Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen. 27 (1980), 127–130, is given for finite rank torsion-free groups. A relationship between nontrivial p-pure subgroups of the additive group of p-adic integers and nontrivial ? [p^?1]-submodules of the field of p-adic numbers is investigated. In particular, a bijective correspondence between these structures is proven using only elementary methods.