# Geometry and Randomness in Group Theory

## May 15-26, 2017

__Beginning Course__

Olga Kharlampovich, Hunter College, CUNY - "Free and Hyperbolic Groups"

Abstract: Geometric Group Theory is built on the ideas and techniques from low dimensional topology, Riemannian geometry, analysis, combinatorics, probability, logic and traditional group theory. The course covers some aspects of geometric, asymptotic, and algorithmic group theory and exciting connections among them. Topics include: free groups, their properties and their subgroups via Stallings subgroup graphs; residual finiteness and its generalizations; groups given by generators and relations; Cayley graphs and the word metric; Van Kampen diagrams and Van Kampen's Theorem; word-hyperbolic groups and algorithmic properties of hyperbolic groups.

Kim Ruane, Tufts University - "Examples of Non-Positively Curved Groups"

Abstract: Hyperbolic (or negatively curved) groups can be characterized in many different ways and essentially all of them turn out to be equivalent to Gromov’s original definition of word hyperbolic groups. There have been many attempts to understand a wider class of non-positively curved groups using different properties -- both algebraic and geometric. In this course, we will discuss some of these properties and give examples of groups which fall into this wider class of non-postively curved groups because they satisfy one or more of these properties. We will also discuss some groups that should perhaps be in this class yet it is unclear how to prove whether they are or not.

__Advanced Course__

Goulnara Arzhantseva, University of Vienna - "Random Groups"

Abstract: There are two distinct aspects of infinite random groups. The first one is inspired by the natural idea to study a typical (or generic) representative of a given class of groups. The second aspect is based on the Erdős probabilistic method and provides exotic random groups, i.e., infinite groups with unusual properties. We will discuss both aspects, beginning with fundamentals such as small cancellation theory and expander graphs.

Tatiana Nagnibeda, University of Geneva - "Amenability"

Abstract: The notion of amenability was introduced by John von Neumann in 1929 in his work on the Banach-Tarski paradox. This very interesting property, defined for a locally compact group, turns out to be related to various areas of mathematics, such as measure theory, representation theory, geometry, probability theory and more -- giving way to a number of equivalent reformulations, some of them conjectural. The course is intended as an introduction into the theory of amenable groups. We will discuss amenability in relation to the algebraic structure of a (finitely generated) group, as well as to such geometric and probabilistic properties as growth, random walks and percolation.

**Prerequisite:**

Undergraduate knowledge in group theory and topology.