Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

The main goal of this thesis is a contribution to the classification of all incidence geometries<p>of rank two on which some group PSL(2,q), q a prime power, acts flag-transitively.<p>Actually we require that the action be RWPRI (residually weakly primitive) and (2T)1<p>(doubly transitive on every residue of rank one). In fact our definition of RWPRI requires<p>the geometry to be firm (each residue of rank one has at least two elements) and RC<p>(residually connected).<p><p>The main goal is achieved in this thesis.<p>It is stated in our "Main Theorem". The proof of this theorem requires more than 60pages.<p><p>Quite surprisingly, our proof in the direction of the main goal uses essentially the classification<p>of all subgroups of PSL(2,q), a famous result provided in Dickson’s book "Linear groups: With an exposition of the Galois field theory", section 260, in which the group is called Linear Fractional Group LF(n, pn).<p><p>Our proof requires to work with all ordered pairs of subgroups up to conjugacy.<p><p>The restrictions such as RWPRI and (2T)1 allow for a complete analysis.<p><p>The geometries obtained in our "Main Theorem" are bipartite graphs; and also locally 2-arc-transitive<p>graphs in the sense of Giudici, Li and Cheryl Praeger. These graphs are interesting in their own right because of<p>the numerous connections they have with other fields of mathematics. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Dans le Chapitre 1 nous étudions les groupes localement compacts lacunaires hyperboliques. Nous caractérisons les groupes ayant un cône asymptotique qui est un arbre réel et dont l'action naturelle est focale. Nous étudions également la structure des groupes lacunaires hyperboliques, et montrons que dans le cas unimodulaire les sous-groupes ne satisfont pas de loi. Nous appliquons au Chapitre 2 les résultats précédents pour résoudre le problème de l'existence de points de coupure dans un cône asymptotique dans le cas des groupes de Lie connexes. Dans le Chapitre 3 nous montrons que le groupe de Neretin est compactement présenté et donnons une borne supérieure sur sa fonction de Dehn. Nous étudions également les propriétés métriques du groupe de Neretin, et prouvons que certains sous-groupes remarquables sont quasi-isométriquement plongés. Nous étudions dans le Chapitre 4 une famille de groupes agissant sur un arbre, et dont l'action locale est prescrite par un groupe de permutations. Nous montrons entre autres que ces groupes ont la propriété (PW), et exhibons des groupes simples au sein de cette famille. Dans le Chapitre 5 nous introduisons l'éventail des relations d'un groupe de type fini, qui est l'ensemble des longueurs des relations non engendrées par des relations plus courtes. Nous établissons un lien entre la simple connexité d'un cône asymptotique et l'éventail des relations du groupe, et donnons une grande classe de groupes dont l'éventail des relations est aussi grand que possible. / In Chapter 1 we investigate the class of locally compact lacunary hyperbolic groups. We characterize locally compact groups having one asymptotic cone that is a real tree and whose natural isometric action is focal. We also study the structure of lacunary hyperbolic groups, and prove that in the unimodular case subgroups cannot satisfy a law. We apply the previous results in Chapter 2 to solve the problem of the existence of cut-points in asymptotic cones for connected Lie groups. In Chapter 3 we prove that Neretin's group is compactly presented and give an upper bound on its Dehn function. We also study metric properties of Neretin's group, and prove that some remarkable subgroups are quasi-isometrically embedded. In Chapter 4 we study a family of groups acting on a tree, and whose local action is prescribed by some permutation group. We prove among other things that these groups have property (PW), and exhibit some simple groups in this family. In Chapter 5 we introduce the relation range of a finitely generated group, which is the set of lengths of relations that are not generated by relations of smaller length. We establish a link between simple connectedness of asymptotic cones and the relation range of the group, and give a large class of groups having a relation range as large as possible.