This habilitation memoir (in French, submitted in May 2014) is made up of five chapters, each being an introduction to work of the author between 2006 and 2014. The core of the memoir consists of the first three chapters, pertaining to geometric group theory. More precisely, it is concerned with large-scale geometry of groups and especially Lie groups (Chapters 1 and 2), and locally compact hyperbolic groups (Chapter 3). Chapter 4 is concerned with the study of groups through their unitary representations and their isometric actions on Hilbert spaces (Kazhdan and Haagerup properties). Finally, Chapter 5 is more concerned with "structural" group theory, studying, for a given group (discrete, or more generally locally compact), the space of its closed subgroups, or closed normal subgroups. A list of open questions is appended. While the memoir mostly surveys published papers, it also presents some unpublished corollaries in Chapter 1 about asymptotic cones of Lie groups.
The goal of this paper is to develop the combinatorics needed to understand in an explicit way the transfer between classical groups and general linear groups
By underlying the commutation relation in a right-angled Coxeter group W, we recover the fact that right-angled Coxeter groups are rigid and we describe the second subgroup of Aut(W) that appears in the decomposition of Aut(W) into a semi-direct product established by J. Tits in "Sur le groupe des automorphismes de certains groupes de Coxeter" (Journal of Algebra 113, (1988), 346-357). ----- En mettant l'accent sur la relation de commutation dans un groupe de Coxeter a angles droits W, on redemontre le fait que les groupes de Coxeter a angles droits sont rigides et on decrit le second sous-gro…
The goal of this paper is to prove how Arthur's results, in the case of split odd orthogonal p-adic groups, imply the Langlands' classification of discrete series. Of course this need the validity of ''fundamental'' lemmas which are not yet available. And this include a Langlands' classification of cuspidal irreducible representations for such groups.
Let G be the set of k-points of a connected semisimple algebraic group of k-rank one over a nonarchimedean local field k. We describe all finitely generated torsion-free discrete subgroups of G\times G acting properly discontinuously and cocompactly on G by left and right multiplication. We prove that after a small deformation in G\times G such a discrete subgroup keeps acting freely, properly discontinuously, and cocompactly on G. ----- Soit G l'ensemble des k-points d'un groupe algebrique semi-simple connexe de k-rang un sur un corps local ultrametrique k. Nous decrivons tous les sous-groupe…
Dans cet article, on montre que l'espace des groupes marqués est un sous-espace fermé d'un ensemble de Cantor dont la dimension de Hausdorff est infinie. On prouve que la dimension de Minkowski de cet espace est infinie en exhibant des sous-ensembles de groupes marqués à petite simplification dont les dimensions de Minkowski sont arbitrairement grandes. On donne une estimation des dimensions de Minkowski de sous-espaces de groupes à un relateur. On démontre enfin que les dimensions de Minkowski du sous-espace des groupes commutatifs marqués et d'un ensemble de Cantor défini par Grigorchuk sont…
This paper deals with the Langlands' classification for discrete series of unitary quasi-split p-adic groups. We show that such a classification follows from Arthur's work on the simple trace formula which we can use now thanks to Laumon-Ngo's proof of the fundamental lemma and Waldspurger's work on the consequences of it on the transfer.
The aim of this work, is to describe fairly explicitly a general Arthur's packet for a classical group. The problem to be solved, here, is the decompostion of induced representations. Following previous work on general linear group, such a description implies the needed transfer for the representations in Arthur's packet.
We give a geometric description of a certain class of epimorphisms between complex reflection groups. We classify these epimorphisms, which can be interpreted as ``morphisms'' between the diagrams symbolizing standard presentations by generators and relations for complex reflection groups and their braid groups.
We describe the nilpotent subgroups of the group Bir(P^2(C)) of birational transformations of the complex projective plane. Let N be a nilpotent subgroup of class k>1; then either each element of N has finite order, or N is virtually metabelian.
A cocompact lattice in a semisimple Lie group $G$ is a discrete subgroup $Γ$ such that the quotient $G/Γ$ is compact. Does such a lattice always contain a surface group, i.e. a subgroup isomorphic to the fundamental group of a compact hyperbolic surface? If so, does it contain surface subgroups close (in a precise quantitative sense) to Fuchsian subgroups of $G$, i.e to discrete subgroups of $G$ contained in a copy of $\operatorname{(P)SL}(2,\mathbf{R})$ in $G$? The case $G=\operatorname{PSL}(2,\mathbf{C})$ corresponds to a famous conjecture of Thurston on 3-dimensional hyperbolic manifolds, a…
This part is made of three sections. In the first section we study the family of polynomials whose roots are 4cos2 kπ, (n \geqslant 3,1 \leqslant k < \frac{n}{2}). We obtain n2 in this manner a family of orthogonal polynomials. This will permit us to study in details all the examples which follow. In the second section, we give technical formulae in order noto repeat calculations. In the third section,we give applications, first when the field K is a sub-field of R (presentations of W(H3) and W(H4)) then in the complex case (study of the complex reflection group G(p, p, n), G24 and G27).
The goal of this survey is to present combinatorial modulus that have been recently used to study quasi-conformal properties of boundaries of hyperbolic groups. First, we will recall well known rigidity results and questions that motivated the introduction of these tools. Then we will define combinatorial modulus and the combinatorial Loewner property that provide new approaches to rigidity questions. Finally, we will describe some applications of these tools through recent results. ----- Le but de ce survol est de présenter les modules combinatoires récemment utilisés pour étudier les proprié…
In this article we give a classification of the sub-groups in PSL(2,Z) and of the conjugacy classes of these sub-groups by the mean of an combinatorial invariant: some trivalent diagrams (dotted or not). We give explicit formulae enabling to count the number of isomorphism classes of these structures and of some of their variations, as function of the number of their arcs. Until now, the counting of non-dotted diagrams was an open problem, for it gives also the number of unrooted combinatorial maps, triangular or general respectively. The article ends with the description of a high performance…
The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed sugroups of the group RxZ, which is not trivial : for example, its fundamental group is uncountable.
We describe the links between group theory and psychology, in particular through the works of Piaget. We show that groups appear universally in his description of children's intelligence, and that the notion of groupoid, which was little considered in psychology, may be fundamental. We study in particular the applicability of group theory concepts to the development of educative games.
Un des résultats fondamentaux de la théorie de Bass-Serre est le théorème suivant : un groupe est amalgamé si et seulement si il agit sur un arbre avec comme domaine fondamental un segment. Dans cet article nous donnons un analogue pour les groupes profinis de ce résultat en utilisant la théorie des prographes de Deschamps et Suarez introduite. A fundamental result of Bass-Serre theory is the following theorem: an abstract group is a free product with amalgamation if and only if it acts on a tree with a segment as fundamental domain. In this article, an analogous result for profinite groups wi…
Let K be the function field of a curve over the complex field. Let X be a homogeneous space of a semisimple linear algebraic group. Strong approximation holds for X outside any finite nonempty set of places of K. Strong approximation fails for tori over K.
The results known for Green-Lazarsfeld sets and solvable or nilpotent quotients of Kaehler groups are extended to the class of (compact Kaehler) geometric orbifolds with finite and integral multiplicities. The proofs are by reduction to the known case of compact Kaehler manifolds.
Following Steinberg, we construct an adjoint quotient for the Vinberg semi-group and a section to this quotient. Then, after Ngô, we show the existence of a regular centralizer on it and use it to compute the affine Springer fibers for groups.
This is a thoroughly revised version of math.AG/0502516v1 (24 Feb. 2005). Let k be a field of characteristic zero. Let Y=G/H, where G is a connected linear algebraic group over k and H is a connected closed k-subgroup of G. Let X be a smooth compactification of Y over k. We prove the conjecture set forward in the previous version : the Galois-lattice given by the geometric Picard group of X is flasque. The previous version had only partial results in this direction. They were obtained at the expense of a long d'etour via local and global fields. We owe the drastic improvement to a suggestion b…
In this paper we construct some packets of representations which have to correspond to relatively general Arthurs packets; this is for any classical group $G$ over a p-adic field $F$. An Arthur's packet correspond to a map $ψ$ from $W_{F} \times SL(2,{\mathbb C}) \times SL(2,{\mathbb C})$ into the $L$-group of $G$. The packets we consider here have the property that the centralizer of $ψ$ in the dual group is a finite groupe. Our construction is a combinatorial one which reduce the study of the representations in such a packet to tempered representation of eventualy smaller groups; in fact we …
- Synthèse des travaux présentés en vue d'une Habilitation à Diriger des Recherches - Synthesis of works presented towards the Habilitation degree This is a summary (in French) of my work in number theory, group theory and combinatorics in the last eight years.
This paper, in French, is a celebration of Max Dehn, and an essay of describing some of his results published in the beginning of the 1910's, and their offspring. It has been written up for a winter school in Les Diablerets, March 7-12, 2010: Geometry, topology and computation in groups, 100 years since Dehn's Decision Problems.
Soit X un espace homogène d'un groupe algébrique linéaire connexe G sur C. Soit x un C-point de X. On désigne par H le stabilisateur de x dans G. On montre qu'on peut définir algébriquement le groupe fondamental topologique π_1(X(C),x), si ce groupe fondamental topologique est abélien. Si Pic(G)=0 et H est connexe ou abélien, on calcule π_1(X(C),x) en termes des groupes de caractères de G et H. En outre, si G et X sont définis sur un corps algébriquement clos de caractéristique p quelconque, on calcule la partie première à p du groupe fondamental étale de X en termes des groupes de caractères …
Let $(A_S,S)$ be an Artin-Tits and $X$ a subset of $S$ ; denote by $A_X$ the subgroup of $A_S$ generated by $X$. When $A_S$ is of spherical type, we prove that the normalizer and the commensurator of $A_X$ in $A_S$ are equal and are the product of $A_X$ by the quasi-centralizer of $A_X$ in $A_S$. Looking to the associated monoids $A_S^+$ and $A_X^+$, we describe the quasi-centralizer of $A_X^+$ in $A_S^+$ thanks to results in Coxeter groups. These two results generalize earlier results of Paris. Finaly, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin-Tits…
We show in this article that, for any group $G$ indecomposable for the free product * and non-isomorphic to $\mathbf{Z}$, the canonical inclusion ${\rm Aut}(G^{*n})\to {\rm Aut}(G^{* n+1})$ induces an isomorphism between the homology groups $H_i$ for $n\geq 2i+2$, as was conjectured by Hatcher and Wahl. In fact we show a little more --- in particular, the result is true for any group $G$ if we replace the automorphism group of the free product by the subgroup of symmetric automorphisms. For this purpose we use constructions and acyclicity results due to McCullough-Miller and Chen-Glover-Jensen…
The mapping class group $Γ$ of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of $Γ$ on this graph to find an explicit non trivial quasimorphism on $Γ$ and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a hyperbolic element of $Γ$ with vanishing stable commutator length. This carries out a program proposed by Danny Calegari.
We study the interpolation group whose elements are suitable pairs of formal power series. This group has a faithful representation into infinite lower triangular matrices and carries thus a natural structure as a Lie group. The matrix exponential between its Lie algebra and its matrix representation gives rise to a function with interesting properties extending the usual exponential function to two variables (which are formal power series) We finish with an application to enumerative combinatorics and the description of an algebra which generalizes the interpolation group.
Inspired by the Bruhat-Tits building of SL$_n$($\mathbb Q_p$), we construct a complete metric space X with an action of the tame automorphism group of the affine space Tame($K^n$). The points in X are certain monomial valuations, and X admits a natural structure of Euclidean CW-complex of dimension n-1. When n = 3, and for K of characteristic zero, we prove that X has non-positive curvature and is simply connected, hence is a CAT(0) space. As an application we obtain the linearizability of finite subgroups in Tame($K^3$).
A connected reductive group G over a field k may be written as a quotient H/S, where the k-group H is an extension of a quasitrivial torus by a simply connected semisimple group, and S is a flasque k-torus, central in H (a flasque torus is a torus whose cocharacter group is an H^1-trivial Galois lattice). The flasque torus S is well-defined up to multiplication by a quasitrivial torus. Such presentations G=H/S lead to a simplified approach of the Galois cohomology of G and of related objects, such as the Brauer group of a smooth compactification of G. When k is a number field, one also recover…
We provide a geometric characterization of manifolds of dimension 3 with fundamental groups of which all conjugacy classes except 1 are infinite, namely of which the von Neumann algebras are factors of type $II_1$: they are essentially the 3-manifolds with infinite fundamental groups on which there does not exist any Seifert fibration. Otherwise said and more precisely, let $M$ be a compact connected 3-manifold and let $Γ$ be its fundamental group, supposed to be infinite and with at least one finite conjugacy class besides 1. If $M$ is orientable, then $Γ$ is the fundamental group of a Seifer…
