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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