Geometric presentations of Lie groups and their Dehn functions

We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local fields, uses and extends Abels’ theory of multiamalgams of graded Lie algebras, in order to provide workable presentations of these groups.     

Any virtually polycyclic group admits a NIL-affine crystallographic action

In this paper, we prove that any virtually polycyclic group admits a properly discontinuous and cocompact action on a simply connected, connected nilpotent Lie group N. This action is affine in the sense that the group acts as a subgroup of the affine group Aff(N)=N?Aut(N) of connection preserving automorphisms of N. As a consequence, we also obtain that any virtually polycyclic group admits a polynomial crystallographic action of degree bounded above by its Hirsch length.     

Distinguished orbits and the L–S category of simply connected compact Lie groups

We show that the Lusternik–Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G.     

Conjugacy Length in Group Extensions

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behavior of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products ?^d ? ?^k have at most exponential (if k > 1) or linear (if k = 1) conjugacy length functions.     

Contributions to Real Exponential Lie Groups

 A Lie group is called exponential if its exponential map is surjective. It is called weakly exponential if the exponential image is dense, which is equivalent to the connectivity of each of the Cartan subgroups (compare [11]). In the present paper the authors study exponential Lie groups which are of mixed type, i.e., neither solvable nor semisimple. Necessary conditions and also, for special mixed Lie groups, sufficient conditions are given for exponentiality. Several counter examples are provided showing that none of the conditions which have surfaced during the course of our investigation can work as necessary and sufficient ones. All conditions considered deal with centralizers of ad-nilpotent elements of the Lie algebra. For example, it is shown that if G is exponential, there is a rather large characteristic subgroup B which contains the nilradical, all Levi factors, and all maximal compactly embedded subgroups, which is also exponential. Moreover, this subgroup is also Mal’cev splittable so that one can apply earlier results on Mal’cev splittable exponential Lie groups, which characterize exponentiality of these Lie groups (also by conditions concerning the centralizers of ad-nilpotent elements). (Received 1 June 1999; in final form 28 December 1999)     

A Cartan–Hadamard Theorem for Banach–Finsler Manifolds

In this paper we study Banach–Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point. In this context we generalize the classical theorem of Cartan–Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach–Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.     

Kazhdan constants for conjugacy classes of compact groups

Kazhdan constants relative to conjugacy classes of compact groups are computed. They depend on the nontrivial irreducible characters of the respective group. The result is applied, in particular, to finite groups of Lie type, symmetric groups, and the group SU(n).     

Valuation theory of exponential Hardy fields I

We describe the valuation theoretic properties of the Hardy fields associated to models of , where T is the theory of a polynomially bounded o-minimal expansion of the reals and is the real exponential function. We deduce that has levels with parameters and is exponentially bounded. We establish a maximality property of , the Hardy field of the expansion by the restricted analytic functions and power functions. Received: 10 July 2000 ; in final form: 15 April 2002/Published online: 24 February 2003 RID="*" ID="*" This paper was written while both authors were partially supported by a Canadian NSERC research grant.     

Tychonoff property for linear groups

A criterion for a wide class of topological groups which includes linear discrete groups and Lie groups to be Tychonoff groups is established. The main result provides a criterion for an almost polycyclic group to have the Tychonoff property. By the well-known Tits alternative, this yields the required criterion for linear discrete groups. In conclusion it is pointed out that a particular case of the presented proof yields a Tychonoff property criterion for Lie groups. In addition, an example of a polycyclic group without Tychonoff subgroups of finite index is constructed. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 269–279, February, 1998. The author wishes to express his gratitude to R. I. Grigorchuk for setting the problem and his interest in the work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00182 and by the American Mathematical Society Fund.     

New properties of lattices in Lie groups

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in $\text{PSL}\text{(2,}\text{R}\text{)}$) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups. To cite this article: F. Grunewald, V. Platonov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Groups with Bounded Chernikov Conjugate Classes of Elements

We consider BCC-groups, that is groups G with Chernikov conjugacy classes in which for every element x G the minimax rank of the divisible part of the Chernikov group G/C _G(x ^G) and the order of the corresponding factor-group are bounded in terms of G only. We prove that a BCC-group has a Chernikov derived subgroup. This fact extends the well-known result due to B. H. Neumann characterizing groups with bounded finite conjugacy classes (BFC-groups).     

On growth,recurrence and the Choquet-Deny Theorem for <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

We first study the growth properties of p-adic Lie groups and its connection with p-adic Lie groups of type R and prove that a non-type R p-adic Lie group has compact neighbourhoods of identity having exponential growth. This is applied to prove the growth dichotomy for a large class of p-adic Lie groups which includes p-adic algebraic groups. We next study p-adic Lie groups that admit recurrent random walks and prove the natural growth conjecture connecting growth and the existence of recurrent random walks, precisely we show that a p-adic Lie group admits a recurrent random walk if and only if it has polynomial growth of degree at most two. We prove this conjecture for some other classes of groups also. We also prove the Choquet-Deny Theorem for compactly generated p-adic Lie groups of polynomial growth and also show that polynomial growth is necessary and sufficient for the validity of the Choquet-Deny for all spread-out probabilities on Zariski-connected p-adic algebraic groups. Counter example is also given to show that certain assumptions made in the main results can not be relaxed.     

The Field of Reals with Multisummable Series and the Exponential Function

We show that the field of real numbers with multisummable realpower series is model complete, o-minimal and polynomially bounded.Further expansion by the exponential function yields again amodel complete and o-minimal structure which is exponentiallybounded, and in which the Gamma function on the positive realline is definable. 2000 Mathematics Subject Classification:primary 03C10, 32B05, 32B20; secondary, 26E05.     

Finite dimensional groups of local diffeomorphisms

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic groups and their subgroups. Such groups are called finitedimensional. We obtain that cyclic groups, virtually polycyclic groups, finitely generated virtually nilpotent groups and connected Lie groups of local biholomorphisms are finite-dimensional. We provide several methods to identify finite-dimensional groups and build examples.As a consequence we generalize results of Arnold, Seigal–Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, including for example virtually polycyclic groups and in particular finitely generated virtually nilpotent groups.     

A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker-Campbell-Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.     

The exponential image of simple complex lie groups of exceptional type

Let G be a connected reductive complex Lie group. Let E _G be the image of the exponential map of G and E' _G its complement in G. We give a purely algebraic characterization of the set E _G and also describe an algorithm for finding all conjugacy classes of G in E' _G . We are mainly interested in the case when the Lie algebra of G is simple and exceptional. Full details are provided for groups G of type G ₂, F ₄, and E ₆. If G is of type G ₂ then there are only two such conjugacy classes.This work was supported by NSERC Grant A-5285.     

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.     

Automorphismes analytiques des domaines produits

In this paper, I study the group of analytic automorphisms of a bounded product domain in the spaceC(S,C) of continuous functions on a compact spaceS. I prove that its automorphism group is a Lie group and I am able to prove which are the bounded symmetric ones.     

Geometric groups. I

We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.