Colloquium to honor the memory of Jacques Tits - ^3rd day

Program

• 09:30 → 10:20 - On the Tits Alternative and its Refinements
  I shall present Tits' original statement and method of proof of his celebrated alternative for finitely generated linear groups and present various refinements that have been obtained over the years as well as open problems. In particular, I shall discuss the notion of strongly dense free subgroup appearing in recent joint work with Guralnick and Larsen on this topic as well as connections to the study of word maps and expander graphs.

  Speaker : Emmanuel BREUILLARD

• 10:30 → 11:20 - Tits Alternative for the Cremona Group
  A group G satisfies the Tits alternative if any subgroup contains either a non-Abelian free subgroup or a solvable group of finite index. This alternative has been proved by Jacques Tits for the linear groups (up to restricting to finitely generated subgroups when the field is of positive characteristic). Since then, this alternative has been proved for many other groups. In particular, Serge Cantat (in the case of finitely generated subgroups) and Christian Urech proved that the Cremona group, namely the group of birational transformations of the projective plane, satisfies the Tits alternative. After an introduction to the Cremona group, the goal of this talk is to give their proof's strategy of this important result.

  Speaker : Anne LONJOU

• 11:30 → 12:20 - Simple Algebraic Groups among Locally Compact Groups
  The real simple Lie groups and the simple algebraic groups over non-Archimedean local fields are ubiquitous in mathematics. For that reason, they are probably the most remarkable members of the class of non-discrete simple locally compact groups. But what makes them so special within that class? What are their characteristic properties as locally compact groups? The goal of this talk is to present an overview of attempts at answering that question, by adopting algebraic, geometric, and representation-theoretic points of view.

  Speaker : Pierre-Emmanuel CAPRACE