Colloque en l'honneur de Gabriel Sabbagh

Paris - Chevaleret, 15 - 16 septembre 2008

Groupes, modules et logique
Groups, modules and logic

Emmanuel Breuillard (Paris 11)
Strong Tits alternative for finitely generated linear groups and arithmetic heights on character varieties

We prove an effective uniform version of the Tits alternative on an arbitrary field and derive several new results on the structure of linear groups, mainly about their growth and number of relations. The main statement can be reformulated in terms of first order logic as the equality between seemingly unrelated algebraic varieties. This allows to reduce mod p and get new applications such as bounds on the girth of Cayley graphs of subgroups of $GL(n,F_q)$.

Proofs rely on the notion of "arithmetic spectral radius" of a finite family of matrices and key ingredients are some results from Diophantine Geometry such as theorems of Bilu and Zhang on the set of points of small height on algebraic varieties.

We will also mention the Lehmer conjecture, in connection with the solvable case, which turns out to be surprisingly harder.

Abderezak Ould Houcine (Lyon I)
Superstable groups acting on trees

We initiate a study of superstable groups acting on trees (simplicial or real). We will show that if a free product with amalgamation G=G_1*_AG_2 is superstable with the index of A is not 2 in G_1, then G interprets a simple superstable non-ω-stable group. We show also that an action of an ω-stable group on a simplicial tree is trivial, and in particular a free product with amalgamation or an HNN-extension is not ω-stable. We study `minimal' superstable groups acting on trees and we give some of their properties.

Elisabeth Bouscaren (CNRS - Paris 11)
Semiabelian varieties over separably closed fields, exact sequences and maximal divisible subgroups.

(Joint with F. Benoist (Leeds/Paris 11) and A. Pillay (Leeds)). Given a separably closed field K of characteristic p>0 and finite degree of imperfection, we study the functor which takes a semiabelian variety G over K, to G^#, the maximal divisible subgroup of G(K) (the group of K-rational points of G). We show that this functor does not always preserve exact sequences. We relate preservation of exactness to issues of descent and to model theoretic properties of G^#.

Variétés semiabéliennes sur un corps séparablement clos, suites exactes et sous-groupes divisibles maximaux.

(Travail en commun avec F. Benoist (Leeds et Paris 11) et A. Pillay (Leeds).) Etant donné un corps séparablement clos K, de caractéristique p>0 et de degré d'imperfection fini, nous étudions le foncteur qui envoie une variété semiabélienne G, définie sur K, sur G^#, le plus gros sous-groupe divisible de G(K) (le groupe des points K-rationnels de G). Nous montrons que ce foncteur ne préserve pas toujours les suites exactes et qu'il y a une relation entre la préservation de l'exactitude, des questions de descente et les propriétés modèle-théoriques du groupe G^#.

Anatole Khélif (IUFM Paris)
Bi-interprétabilité et structures QFA : étude des groupes résolubles et des anneaux commutatifs

Une structure S de type fini est dite QFA (pour quasi finiment axiomatisable) s'il existe un énoncé du premier ordre satisfait par S telle que toute structure de type fini qui la satisfait est isomorphe à S. Une structure S est dite première si elle se plonge élémentairement dans tout modèle de sa théorie, c'est à dire si elle est un modèle premier de sa théorie. Nous étudions le cas de quelques groupes résolubles et des anneaux commutatifs.

John Wilson (Oxford)
The soluble radical of a finite group

Although solubility for finite groups is defined in terms of the existence of a series of normal subgroups of a certain type, solubility can also be characterized in ways that seem quite unrelated to series and that shed a different light on the significance of solubility. We shall discuss some of these characterizations.   Each finite group G has a unique largest soluble normal subgroup (called its radical). In further attempts to understand what it means for  G to be soluble we shall discuss some characterizations of the elements of the radical of G. 

Thierry Coulbois (Marseille III)
Dynamique des automorphismes du groupe libre

Je commencerai par introduire le groupe des automorphismes (extérieurs) d'un groupe libre. Je montrerai qu'il agit sur un espace intéressant : l'Outre espace qui est constitué d'arbres simpliciaux et que la dynamique de l'action sur cet espace donne des informations intéressantes sur Out(Fn). Je proposerai ensuite une parenthèse montrant une similitude avec l'action du groupe des classes modulaires sur l'espace de Teichmüller. Je parlerai ensuite de la compactification de l'Outre espace par les arbres réels et de la dynamique de l'action de Out(Fn) sur ce bord. Le but sera d'arriver à parler de nos travaux sur les laminations et les systèmes d'isométries.

Durant tout l'exposé j'essayerai de proposer des exemples concrets d'automorphismes pour montrer que tout cela est finalement assez combinatoire.

Dugald Macpherson (Leeds)
Pseudofinite groups and permutation groups

I will discuss a collection of topics around pseudofinite groups (infinite models of the theory of finite groups). These include: virtual solubility of stable pseudofinite groups (joint with Tent); results of Ryten on the model theory of finite simple groups, and applications to primitive pseudofinite permutation groups (work with Liebeck and Tent); some observations with Elwes and Ryten on the structure of pseudofinite groups with supersimple theory.

Francis Oger (CNRS - Paris 7)
Elementary equivalence between polycyclic groups

In a correspondence with R.B. Warfield thirty years ago, G. Sabbagh suggested that finitely generated nilpotent groups, and more generally polycyclic groups, can be elementarily equivalent without being isomorphic. Later on, we gave algebraic characterizations of that situation, using profinite completions and the cancellation properties of the infinite cyclic groups in direct products. More recently, we obtained new results, using the notion of quasi-finitely axiomatizable (QFA) group introduced by A. Nies.

Wildrid Hodges (Queen Mary, London)
Abelian groups: algebraic versus model-theoretic methods

Gabriel Sabbagh is one of a small group of people who in the early 1970s invented the model theory of modules, which greatly generalised what model theorists already knew about abelian groups. With a third of a century's hindsight I describe the character of this theory - for example how it broke free from some of the aims of Tarski's earlier work, and how in its early days it stopped short of fully embracing the techniques of stability theory. I illustrate with the case of relatively categorical pairs of abelian groups.

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