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# 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:**

**December 2 ** Yash Lodha, Lecture 4 : Left orderability and local indicability.

[On zoom : meeting id = 898 2923 9327 Password: 993204]

Abstract: We shall prove that locally indicable groups are left orderable, and also discuss counterexamples that witness the failure of the converse. We shall prove the striking theorem of Witte-Morris which states that amenable left orderable groups are locally indicable.

**History:**

**November 25** Yash Lodha, Lecture 3 : Thompson's groups and left orderability.

[On zoom : meeting id = 898 2923 9327 Password: 993204]

In the first half, we shall continue our study of Thompson's groups. We shall define the groups T,V and prove that they are simple. In the second half, we shall study the notion of left orderability and define the space of left orders. We shall prove that for countable groups left orderability is equivalent to admitting faithful actions by orientation preserving homeomorphisms of the real line. We shall also study the notions of local indicability and C-orderability, and demonstrate that they coincide.

**November 18** Yash Lodha : Lecture 2 on Thompson's groups F,T and V.

In this lecture we shall prove that F does not contain nonabelian free subgroups, and yet does not satisfy a law. We shall state and prove the 2-chain Lemma. Next, we shall define the groups T,V and demonstrate that they are simple. We shall also discuss finiteness properties of these groups. If time permits, we shall explain why the amenability question for F is interesting and why F is not elementary amenable.

**November 11** Yash Lodha : Lecture 1 on the Thompson's groups F,T and V.

This is the first of a series of lectures on Thompson Group. In the first lecture we shall provide a basic introduction to Thompson's groups F,T and V. We shall start with the definitions, and study the basic algebraic properties of the groups including normal subgroup structure and simplicity. We shall study the standard actions of the groups on the real line, circle and the cantor set.

**November 4 **Gonzalo Ruizstolowicz

We will first review and clarify a few points from the lecture of October 14. Then we will proceed to the proof of Gromov's theorem along the following steps :

- Proof by Ozawa that polynomial growth implies slow entropy growth

- This implies that any finitely generated group of polynomial growth has Shalom´s property Hfd.

- It then follows that the group has a finite index subgroup with an epimorphism to the integers.

- By induction, any group of polynomial growth is virtually polycyclic.

- Finally Wolf´s theorem about polycyclic groups and growth, implies Gromov´s theorem.

**October 21 and 28: **Kathryn Hess (on Zoom https://epfl.zoom.us/j/81814966472 )

* Introduction to the braid group*

Definitions of the braid group as isometry classes of braids and as a fundamental group. Presentation by generators and relation with the symmetric group. The Artin representation theorem, the associated fibration of configuration spaces.

Here are the links related to K.Hess Lectures :

- Lecture notes October 21
- Lecture notes October 28
- Video recording October 21
- Video recording October 28

**October 14: **Gonzalo Ruizstolowicz

Gromov's Theorem on nilpotent group states that * any finitely generated group with polynomial growth contains a nilpotent subgroup of finite index. *In 2015, Ozawa has given a proof based on (reduced) cohomology of a group with coefficients in a unitary representation. In this talk, we will present the Harmonic Analysis prerequisites for the Shalom & Ozawa proof of the Gromov´s Polynomial Growth Theorem. We are going to study how it is possible, with these tools, to reduce the problem to a linear group of polynomial growth.

**October 7: **Marc Troyanov

In this lecture I will give some applications of the Milnor-Svarc Lemma, in particuler I will discuss the relation with the growth of groups and prove the Milnor-Svarc Theorem in Riemannian geometry. I will also discuss some other quasi-isometry invariants.

**September 30 : **Marc Troyanov:

This lecture will be the continuation of the previous one. I will pursue the discussion of quasi-isometry. I will discuss group actions on proper length metric spaces and give a proof of the *Milnor-Svarc Lemma* and discuss some quasi-isometry invariants.

**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.

**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, moduli 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.