On the structure of finite coverings of compact connected groups

Finite-sheeted covering mappings onto compact connected groups are studied. We show that for a covering mapping from a connected Hausdorff topological space onto a compact (in general, non-abelian) group there exists a topological group structure on the covering space such that the mapping becomes a homomorphism of groups. To prove this fact we construct an inverse system of covering mappings onto Lie groups which approximates the given covering mapping. As an application, it is shown that a covering mapping onto a compact connected abelian group G must be a homeomorphism provided that the character group of G admits division by degree of the mapping. We also get a criterion for triviality of coverings in terms of means and prove that each finite covering of G is equivalent to a polynomial covering.     

Rigidity of group actions on solvable Lie groups

We establish analogs of the three Bieberbach theorems for a lattice in a semidirect product where is a connected, simply connected solvable Lie group and is a compact subgroup of its automorphism group. We first prove that the action of on is metrically equivalent to an action of on a supersolvable Lie group. The latter is shown to be determined by itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group as a lattice in a semidirect product with being a finite group whose order is bounded by a constant only depending on the dimension of . This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup that the quotient is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group. Received: 27 August 1999     

Cocycles Over Abelian TNS Actions

We study extensions of higher-rank Abelian TNS actions (i.e. hyperbolic and with a special structure of the stable distributions) by compact connected Lie groups. We show that up to a constant, there are only finitely many cohomology classes. We also show the existence of cocycles over higher-rank Abelian TNS actions that are not cohomologous to constant cocycles. This is in contrast to earlier results, showing that real valued cocycles, or small Lie group valued cocycles, over higher-rank Abelian actions are cohomologous to constants.     

Characters and coverings of compact groups

We consider characters and finite-sheeted coverings of compact connected Abelian groups and prove analytic and algebraic properties of characters. As an application of these results, we show that the character group of a compact connected Abelian group with trivial finite-sheeted coverings is divisible.     

On Covering Groups

We study finite-sheeted covering mappings from compact connected spaces onto compact connected groups. For such a covering mapping we consider a method of supplying its covering space by a group structure. The covering space endowed with that group structure is a topological group, and a given covering mapping turns into a homomorphism of compact connected groups.

     

Connected finite loop spaces with maximal tori

Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence for this conjecture by showing that the associated action of the Weyl group on the maximal torus always represents the Weyl group as a crystallographic group. We also develop the notion of normalizers of maximal tori for connected finite loop spaces, and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal torus is equivalent to a compact connected Lie group if it has the right normalizer of the maximal torus. Actually, in the cases under consideration the information about the Weyl group is sufficient to give the answer. All this is done by first studying the analogous local problems.

     

A decomposition theorem for compact groups with an application to supercompactness

We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group.     

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hähl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

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.

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Herrn Professor Dr. Eelmut Karzel zum 60. Geburtstag     

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.     

Groups with T1 primitive ideal spaces

We give a simple necessary and sufficient condition for the group C¹-algebra of a connected locally compact group to have a T₁ primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C¹-algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C¹-algebra of a discrete finitely generated solvable group is T₁ if and only if the group is a finite extension of a nilpotent group.     

On invariant sets in Lagrangian graphs

In this exposition, we show that the Hamiltonian is always constant on a compact invariant connected subset which lies in a Lagrangian graph provided that the Hamiltonian and the graph are sufficiently smooth. We also provide some counterexamples to show that if the Hamiltonian function is not smooth enough, then it may be non-constant on a compact invariant connected subset which lies in a Lagrangian graph.     

Central Fourier analysis for Lorentz spaces on compact Lie groups

We determine the critical indexes for theL ^p,q uniform boundedness of the characters of a compact connected semisimple Lie group. These results show a basic difference between the simple and the semisimple case, a situation which does not appear when dealing withL ^p spaces. We provide applications to the general theory of central functions on compact Lie groups.     

Compact Three-Dimensional Elation Quadrangles

We show that each compact three-dimensional EGQ is isomorphic either to the real symplectic quadrangle, or to a translation quadrangle of Tits type. In particular, the elation group is one of the two classical elation groups, and thus a simply connected nilpotent Lie group.     

A splitting result for compact symplectic manifolds

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group K in a Hamiltonian fashion. We prove that the squared moment map ∥μ∥² is constant if and only if K is semisimple and the manifold is K-equivariantly symplectomorphic to a product of a flag manifold and a compact symplectic manifold which is acted on trivially by K. In the almost-Kähler setting the symplectomorphism turns out to be an isometry.     

Amenability properties of the centres of group algebras

Let G be a locally compact group, and ZL¹(G) be the centre of its group algebra. We show that when G is compact ZL¹(G) is not amenable when G is either non-abelian and connected, or is a product of infinitely many finite non-abelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL¹(G).     

Geometrizing Infinite Dimensional Locally Compact Groups

We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.