Model theory can express properties of algebraic subsets of complex n-space. The constructible subsets are precisely the first order definable subsets, and varieties correspond to maximal consistent collections of formulas, called types. Moreover, the topological dimension of a constructible set is equal to the Morley rank of the formula which defines it.

Model theory can express properties of algebraic subsets of complex n-space. The constructible subsets are precisely the first order definable subsets, and varieties correspond to maximal consistent collections of formulas, called types. Moreover, the topological dimension of a constructible set is equal to the Morley rank of the formula which defines it.

This thesis is devoted to the analysis of relations between two major conjectures in the theory of groups of finite Morley rank. One of them is the Cherlin-Zil&#039 / ber Algebraicity Conjecture which states that infinite simple groups of finite Morley rank are isomorphic to simple algebraic groups over algebraically closed fields. The other conjecture is due to Hrushovski and it states that a generic automorphism of a simple group of finite Morley rank has pseudofinite group of fixed points. Hrushovski showed that the Cherlin-Zil&#039 / ber Conjecture implies his conjecture. Proving his Conjecture and reversing the implication would provide a new efficient approach to prove the Cherlin-Zil&#039 / ber Conjecture. This thesis proposes an approach to derive a proof of the Cherlin-Zil&#039 / ber Conjecture from Hrushovski&#039 / s Conjecture and contains a proof of a step in that direction. Firstly, we show that John S. Wilson&#039 / s classification theorem for simple pseudofinite groups can be adapted for definably simple non-abelian pseudofinite groups of finite centralizer dimension. Combining this result with recent related developments, we identify definably simple non-abelian pseudofinite groups with Chevalley or twisted Chevalley groups over pseudofinite fields. After that in the context of Hrushovski&#039 / s Conjecture, in a purely algebraic set-up, we show that the pseudofinite group of fixed points of a generic automorphism is actually an extension of a Chevalley group or a twisted Chevalley group over a pseudofinite field.

Notre travail ici concerne certaines pistes pour des constructions nouveaux groupes, et en particulier de contre-exemples à la conjecture de Cherlin-Zilber. On parvient à trouver une réponse pour la stabilité de groupes CSA existentiellement clos. On exhibe un mot de groupe en deux variables qui a la propriété d’indépendance par rapport à la classe de groupes hyperboliques sans torsion. On en déduit que l’équation correspondante donne la propriété d’indépendance des groupes CSA existentiellement clos, ce qui en particulier implique leur instabilité. En outre, on prouve que les équations, et en particulier les ensembles définissables sans quantificateurs, définissent des ensembles stables dans les boules bornées des produits libres de groupes, en utilisant la version finie du théorème de Ramsey. Enfin, on introduit certains groupes construits comme tours particulières de produits libres et d’extensions HNN. / Our work here relates to certain routes for the construction of new groups, and in particular, of counter-examples to the Cherlin-Zilber conjecture. We managed to find an answer for the stability of existentially closed CSA- groups. We build a group word in two variables that has the independence property relatively to the class of torsion-free hyperbolic groups. We deduced that the corresponding equation gives the independence property of existentially closed CSA groups which in turn implies their instability. Moreover, we demonstrate that group words, and in particular quantifierfree definable sets, define stable sets in bounded balls of free products of groups using a finite version of Ramsey’s theorem. Finally, we introduce certain groups constructed as special towers of free products and HNN-extensions.

Cette thèse traite essentiellement des liens qui peuvent exister entreles groupes de rang de Morley fini et les groupes algébriques linéaires. Eneffet, nous y établissons quelques propriétés algébriques aux K-groupes ;d'ailleurs une étude de linéarité sur ces groupes est dressée et permeten particulier d'obtenir une généralisation du théorème de Levi sur ladécomposition des groupes algébriques. Ensuite, nous étudions dans ununivers de rang de Morley fini, une action définissable de SL2(K) surun groupe abélien SL2(K)-minimal V où K est un corps définissable decaractéristique positive p > 0. À cet effet, nous montrons que le rang deMorley rk(V ) de V est pair et multiple de rk(K). Enfin, nous analysonssous quelles conditions, étant donné G un groupe algébrique sur un corpsalgébriquement clos de caractéristique non nulle, le quotient G=Z(G) estdéfinissablement linéaire.Par ailleurs, nous montrons sous certaines hypothèses le groupe desautomorphismes définissables d'un K*-groupe simple est interprétable. / This thesis essentially focuses on relationships that may exist betweengroups of finite Morley rank and linear algebraic groups. Indeed, weestablish some algebraic properties to K-groups; while a linearity studyon these groups is drawn and allows in particular to obtain an analogueto Levi decomposition theorem of algebraic groups. Next, in a univers offinite Morley rank, we study a definable action of SL2(K) on an abeliangroup V such as V is SL2(K)-minimal, where K is an definable field ofnonzero characteristic. For that purpose, we show that Morley rank ofV denoted rk(V ) is even and multiple of rk(K). Finally, we analyze theconditions under which, given an algebraic group G over an algebraicallyfield of nonzero characteristic, the quotient G=Z(G) is definably linear.Besides, we show under certain assymptions that the group of definable automorphism of a simple K*-group is interpretable.