**Geometric Group Theory**

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

We strongly recommend that graduate students have already had the equivalent of rigorous first year graduate-level courses in topology, algebra and analysis.

The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

**COURSE TITLES AND DESCRIPTIONS:**

There will be 9 week-long minicourses aimed at graduate students, with one lecture a day, as follows.

**Week 1:**

Pierre-Emmanuel Caprace: Structure of CAT(0) spaces and their isometry groups

Tsachik Gelander: Arithmetic Groups, Locally Symmetric Manifolds and some Asymptotic Invariants

Michael Kapovich: Quasi-isometric rigidity

**Week 2:**

Michah Sageev: CAT(0) cube complexes

Amie Wilkinson: Geometric rigidity and the geodesic flow in negative curvature

Dave Morris: Some arithmetic groups that do not act on the circle

**Week 3:**

Mladen Bestvina: Topology and Geometry of Outer space

Emmanuel Breuillard: Property T, expanders and approximate groups

Vincent Guirardel: Rotating families, Dehn fillings and small cancellation

**
Mladen Bestvina: Topology and Geometry of Outer space**

This will be a hands-on introduction to Outer space. I will emphasize analogies with Teichmüller space throughout, although prior knowledge of it is not assumed. I will give ideas of proofs (and references to full details). Possible breakdown by lectures is the following:

Definition of Outer space, Teichmüller space, spine and basic properties. Group-theoretic consequences.

Lipschitz distance, optimal maps, train tracks. Comparison with Teichmüller's theorem.

Classification of automorphisms of free groups. Axes.

Boundary of Outer space. Dynamics.

Folding paths. Negatively curved features.

**
Emmanuel Breuillard: Property T, expanders and approximate groups**

In this minicourse I will study the geometry of Cayley graphs of finite and infinite groups through various quantities such as their growth, isoperimetry, girth, diameter, expansion, etc. While this is a huge subject at the heart of geometric group theory, I will put a special emphasis in these lectures on the notion of expander graph, which arises in various geometrical contexts in connection with property T and property tau. One theme of the course will be to show how to go back and forth between finite and infinite groups, while our main goal will be to give an introduction to the so-called approximate groups and the related new methods, partly based on combinatorics, for establishing that certain Cayley graphs of finite simple groups are expanders.

**
Pierre-Emmanuel Caprace: Structure of CAT(0) spaces and their isometry groups**

The study of spaces and groups of non-positive curvature is one of the foundational topics in geometric group theory. The category of CAT(0) spaces provides a unified framework to study Riemannian manifolds of non-positive sectional curvature, Euclidean and non-Euclidean buildings, as well as many other non-positively curved cell complexes. The aim of this minicourse is to present recent results on the structure of CAT(0) spaces and groups, based on the study of the full isometry group of a proper CAT(0) space, viewed as a locally compact group. The main goals are to generalize classical results (eg. Borel density) to CAT(0) spaces, and to highlight new characterizations that single out symmetric spaces and Euclidean buildings amongst all proper CAT(0) spaces. This is based on joint work with Nicolas Monod.

**
Tsachik Gelander: Arithmetic Groups, Locally Symmetric Manifolds and some Asymptotic Invariants**

I will start with a brief introduction to the theory of arithmetic groups and locally symmetric spaces. After discussing basic properties, I will explain some of the classical results, such us Borel density theorem, Kazhdan--Margulis theorem, finite presentability, etc. If time allows I will also give an overview about rigidity theory. The 2nd half of the course will be devoted to the study of asymptotic invariants (of topological and representation theoretical nature) of $G/Gamma$ when the volume tends to infinity. For instance I will explain the new approach of Local Convergence of manifolds and convergence of Invariant Random Subgroups which in particular implies a strong version of the classical Lueck approximation theorem.

**
Vincent Guirardel: Rotating families, Dehn fillings and small cancellation**

Given a group acting on a hyperbolic space, there is a notion of small cancellation for a subgroup H<G. This notion applies for instance to the cyclic group generated by a large power of a pseudo-Anosov element of the mapping class group, or of an iwip automorphism of the free group. This also applies to some subgroups of parabolic subgroups of a relatively hyperbolic groups.

Following Delzant-Gromov, one can construct from such a small cancellation subgroup a rotating family on a coned-off space Y. Such a rotating family consists of a discrete subset C\subset Y, together with a subgroup G_c<G fixing c for each c\in C, and such that this data is G-invariant. Under natural hypotheses saying roughly that the points in C are far enough from each other, and that non-trivial elements of G_c "rotate by a large angle", the structure of the (normal) subgroup N generated by the groups G_c is a free product, and the quotient space Y/N is still hyperbolic, and its geometry is controlled.

This allows for instance to describe Dehn fillings of relatively hyperbolic groups (following Groves-Manning and Osin), and the normal subgroup of a large power of a pseudo-Anosov element. I will also give an application to the isomorphism problem for relatively hyperbolic group.

**
Michael Kapovich: Quasi-isometric rigidity**

One of the key questions of the geometric group theory is interaction between algebraic and geometric properties of groups. Loosely speaking, a group is called quasi-isometrically rigid if one can essentially recover its algebraic structure from its geometric structure. Quasi-isometries and commensurations are the technical tools to formalize the above concept. In this minicourse I will introduce some tools for proving quasi-isometric rigidity of groups and prove Mostow rigidity theorem for discrete cocompact isometry groups of hyperbolic space and Tukia's theorem on quasi-isometric rigidity for such groups.

**
Dave Morris: Some arithmetic groups that do not act on the circle**

The group SL(3,Z) cannot act (nontrivially) on the circle (by homeomorphisms). We will see that many other arithmetic groups also cannot act on the circle. The discussion will involve several important topics in group theory, such as amenability, Kazhdan's property T, ordered groups, bounded generation, and bounded cohomology.

**
Michah Sageev: CAT(0) Cube Complexes**

CAT(0) cube complexes are a special class of CAT(0) spaces. On the one hand, Gromov's flag condition provides an easy way to check that such spaces are CAT(0) and by now we know that there are many natural classes of groups which act on such spaces. On the other hand, CAT(0) cube complexes exhibit a rich combinatorial structure due to the existence of hyperplanes. Thus, much more is known about groups that act on such spaces than about groups that act on CAT(0) spaces in general. We will discuss the basic structure of cube complexes, including Gromov's flag condition, hyperplanes, halfspaces and generalized vertices. We will then go on to discuss pocsets and the duality between pocsets and cube complexes. We will discuss applications to constructing cubical structures on various classes of groups such as Coxeter groups and small cancellation groups, as well as the connection to Kazhdan's property T. We will then move on to discuss intervals, products and essential cores with an eye towards rank rigidity and the the connection between the Roller boundary of spaces and the Poisson boundary of groups which act on them.

**
Amie Wilkinson: Geometric rigidity and the geodesic flow in negative curvature**

The course will focus on the interplay between the geometric properties of a negatively curved manifold M and the dynamical properties of the geodesic flow on its unit tangent bundle T^1M. This is a rich subject with many deep results, and I will concentrate on one: the proof of marked length spectrum rigidity for negatively curved surfaces.

Let S be a closed, negatively curved surface, and let $\rho_S\colon \pi_1(S)\to \RR$ be the {\em marked length spectrum of $S$}: the function $\rho_S$ assigns to each homotopy class the length of the unique closed geodesic in that class. In this course I will present a proof of the following theorem (proved by Otal, and slightly later by Croke): If S, S' are negatively curved manifolds with the same marked length spectrum, then they are isometric. Whether this result extends to dimensions 3 and higher is an open question.

As background for the proof, I will discuss several well-known properties of the geodesic flow in negative curvature, such as existence of a dense orbit, density of periodic orbits, and ergodicity.