Integrability of induction cocycles for Kac-Moody groups

We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is L^p for any p ∈ [1;+∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody lattices.Prépublication de l’Institut Fourier nº 637 (2004); e-mail: http://www-fourier.ujf-grenoble.fr/prepublicatons.html     

Isomorphisms of Kac-Moody groups which preserve bounded subgroups

A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4.     

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups.     

Analogues of Cayley graphs for topological groups

We define for a compactly generated totally disconnected locally compact group a graph, called a rough Cayley graph, that is a quasi-isometry invariant of the group. This graph carries information about the group structure in an analogous way to the ordinary Cayley graph for a finitely generated group. With this construction the machinery of geometric group theory can be applied to topological groups. This is illustrated by a study of groups where the rough Cayley graph has more than one end and a study of groups where the rough Cayley graph has polynomial growth. Supported by project J2245 of the Austrian Science Fund (FWF) and be an IEF Marie Curie Fellowship of the Commission of the European Union.     

Almost convex groups and the eight geometries

IfM is a closed Nil geometry 3-manifold then ₁(M) is almost convex with respect to a fairly simple geometric generating set. IfG is a central extension or a extension of a word hyperbolic group, thenG is also almost convex with respect to some generating set. Combining these with previously known results shows that ifM is a closed 3-manifold with one of Thurston's eight geometries, ₁(M) is almost convex with respect to some generating set if and only if the geometry in question is not Sol.     

Subgroup distortion in wreath products of cyclic groups

We study the effects of subgroup distortion in the wreath products , where A is finitely generated abelian. We show that every finitely generated subgroup of has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of having distortion function equivalent to the given polynomial. Also, a formula for the length of elements in arbitrary wreath product easily shows that the group has distorted subgroups, while the lamplighter group has no distorted (finitely generated) subgroups. In the course of the proof, we introduce a notion of distortion for polynomials. We are able to compute the distortion of any polynomial in one variable over Z,R or C.     

On the basic algebraic operations in solvable Lie groups

Summary For a simply connected solvable Lie group we specify the structure of the product, the inverse and the exponential map expressed in suitable coordinates (canonical coordinates of the second kind), and point out that in these coordinates the product and inverse are expressed entirely in terms of polynomials, exponential functions and trigonometric functions. We devise algorithms for computing the product, the inverse and the exponential map.     

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that and let Γ be a uniform lattice in G.

(a)

If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.

(b)

If CH fails, then Γ has 2^2ω asymptotic cones up to homeomorphism.

     

Homology stability for classical regular semisimple varieties

We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces, to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology of its associated complex discriminant variety. Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000     

Convex vector lattices and l-algebras

For A an Archimedean Riesz space (=vector lattice) with distinguished positive weak unit e_A, we have the Yosida representation  as a Riesz space in D(X_A), the lattice of extended real valued functions on the space of e_A-maximal ideas. This note is about those A for which  is a convex subset of D(X_A); we call such A “convex”.Convex Riesz spaces arise from the general issue of embedding as a Riesz ideal, from consideration of uniform- and order-completeness, and from some problems involving comparison of maximal ideal spaces (which we won't discuss here; see [10]).The main results here are: (2.4) A is convex iff A is contained as a Riesz ideal in a uniformly complete Φ-algebra B with identity e_A. (3.1) Any A has a convex reflection (i.e., embeds into a convex B with a universal mapping property for Riesz homomorphisms; moreover, the embedding is epic and large).     

Hyperbolic Coxeter groups of large dimension

We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinbergs theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension n, then n 61.Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular their existence in high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means.     

Levi-flat and solvable CR-structures on real Lie-algebras

Let g ₀ be a real Lie-algebra and g its complexification. The main result of the paper is the complete classification of CR-structures in two nontrivial cases: the solvable and the Levi-flat ones. These classifications are obtained via a new Lie-product (depending on J) on the linear subspace p giving the CR-structure.     

Compact spaces and distributive lattices

This note presents a general construction connecting compact locales and distributive lattices, that allows us to reduce results about compactness of locales to theorems about distributive lattices. Two applications are given. One noteworthy feature of our arguments is that they can be formulated both in topos theory and in a predicative theory such as CZF.     

Normalisers in limit groups

Let be a limit group, a non-trivial subgroup, and N the normaliser of S. If has finite -dimension, then S is finitely generated and either N/S is finite or N is abelian. This result has applications to the study of subdirect products of limit groups.Supported in part by Franco-British Alliance project PN 05.004.     

On topologizable and non-topologizable groups

A group G   is called hereditarily non-topologizable if, for every H?G$H?G$, no quotient of H admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked k-generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.     

Classifying spaces of Kač-Moody groups

We study the structure of classifying spaces of Kač-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes'T functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kač-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kač-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kač-Moody groups, and centralizers of finite p-subgroups. Received: 15 June 2000 / in final form: 20 September 2001 / Published online: 29 April 2002     

Some basic properties of planar Singer groups

A planar Singer group is a collineation group of a finite (in this article) projective plane acting regularly on the points of the plane. Theorem 1 gives a characterization of abelian planar Singer groups. This leads to a necessary and sufficient condition for an inner automorphism to be a multiplier. The Sylow 2-structure of a multiplier group and some of its consequences are given in Theorem 3. One important result in studying multipliers of an abelian Singer group is the existence of a common fixed line. We extend this to an arbitrary planar Singer group in Theorem 4. Theorem 5 studies the order of an abelian group of multiplers. If this order equals to the order of the plane plus 1, then the number of points of the plane is a prime. If this order is odd, then it is at most the planar order plus 1.Partially supported by a NSA grant.     

Discrete approximations for complex Kac–Moody groups

We construct a map from the classifying space of a discrete Kac–Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac–Moody group and prove that it is a homology equivalence at primes q different from p. This generalizes a classical result of Quillen, Friedlander and Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac–Moody groups compatible with the Frobenius homomorphism. Our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete Kac–Moody groups.