Topics in geometric group theory.

Doctoral course on Geometric Group Theory

Schedule :       Wednesday  14:15 to 16:00, room CM012   (starting September 23)

Lecturers : Yash Lodha, Christian Urech, Marc Troyanov. The participants are also encouraged to deliver one or several seminar talks.

Goal:  This course/seminar aims at introducing the students to some contemporary aspects of geometric group theory. 

Program:

September 23 : Marc Troyanov: 

    The geometry of finitely generated groups and quasi-isometry invariants.

In this first lecture we will introduce some of the main ideas of geometric group theory, in particular the word metric the Cayley graph and the notion of quasi-isometry. We will give examples  of quasi-isomery invariants and prove the so called Milnor-Svarc Lemma.

September 30 :

 

 

 

 

 


Content: The general philosophy of this subject is to associate to a finitely group a geometric object (namely its Cayley graph) and to investigate the relation between the algebraic and the geometric properties of the group. Some of these relations are very deep and have   
 non trivial algebraic and/or geometric consequences. A special emphasis will be devoted to Artin's groups and the Thompson Group.
 
 The following is a tentative plan of the course, which we shall adapt to the background and taste of the participants :
 
  1. Basic concepts and a few striking though classical results.
  1a. Finitely generated group as a geometric object: the word metric and the Cayley graph.
  1b. The notion of quasi-isometry and example of quasi-isometry invariants
  1b. Growth of group. The Milnor-Svarc Theorem and Gromov's Theorem on virtually nilpotent groups.
  1c. The notion of Gromov Hyperbolic groups.
 
   2. Braid Groups and Artin Group
   2a. Defintions and presentation by generators and relations.
   2b. The braid group and configuration spaces.
   2c. Applications to Riemann surfaces, moduly spaces and monodromy.
   
   3. The Thompson group
   3c.  Finite and infinite presentations, normal forms for elements.
   3d. Simplicity and normal subgroup structure.
   3e. Subgroup structure, the Brin-Squier theorem and distortion.
   3f.  Finiteness properties, the Brown-Geoghegan theorem and BNS invariants.
   3g. Dehn functions.
   3h Fordham's algorithm for computing word length.
   3i. Groups of piecewise projective homeomorphisms.