# Lecture Series November

## Small Cancellation Theory

Lecture Series XII: Small cancellation theory and the Burnside Problem

Speaker: Katrin Tent, Universität Münster

Dates:

(1) Tuesday, November 27, 11 a.m.

(2) Thursday, November 29, 11 a.m.

Abstract:

In these lectures I will report on our recent application of small cancellation theory to obtain a much smaller n such that all Burnside groups of odd exponent at least n are infinite. This is joint work with Atkarskaya and Rips.

Lecture Series XI: Practical generalisations of small cancellation theory

Speaker: Colva Roney-Dougal, University of St Andrews

Dates:

(1) Monday, November 26, 11 a.m.

(2) Wednesday, November 28, 11 a.m.

(3) Friday, November 30, 11 a.m.

Abstract:

Small cancellation theory is a classical technique for analysing van Kampen diagrams over finitely-presented groups, and hence solving a variety of decision problems.

The first lecture will give an introduction to van Kampen diagrams, classical small cancellation theory and the theory of word-hyperbolic groups.

In the final two lectures I'll talk about a long-running project which seeks to generalise small cancellation theory to work over a much larger class of finite presentations, and in particular to solve the word and (more recently) the conjugacy problem over them.

Lecture 2 will cover Stalling's pregroups, and a new class of van Kampen diagrams that we have defined over them which help to expose the large-scale geometry of certain quotients of virtually finite groups.

Lecture 3 will then bring these two strands together: I will present new, polynomial-time procedures to show that a group is hyperbolic and to solve the word problem in linear time.

## Groups of ﬁnite Morley rank

Lecture Series X: A short course on groups of finite Morley rank

Speaker: Joshua Wiscons, California State University, Sacramento

Dates: **(1) Monday, November 19, 11 a.m. **also discussion session at 3 p.m.

(2) Tuesday, November 20, 11 a.m. also discussion session at 3 p.m.

(3) Wednesday, November 21, 11 a.m.

(4) Thursday, November 22, 11 a.m.

Abstract:

This series of four lectures aims to provide an introduction to the study of groups of finite Morley rank (fMr), proceeding from first principles to a sampling of the current state of affairs. No prior knowledge of model theory is necessary, but familiarity with basic group theory will be assumed.

The first two lectures will focus on the general theory of groups of fMr, highlighting the similarities with (and inspiration from) both finite group theory and the theory of algebraic groups. In particular, the Algebraicity Conjecture will be discussed, including both progress and potential obstructions. This conjecture, which posits that every infinite simple group of fMr is isomorphic to an algebraic group, has framed the majority of the research on groups of fMr to date. The third lecture will focus on recent work and guiding problems for permutation groups of fMr; the fourth will touch on a variety of other active topics such as geometries of fMr, modules of fMr, and small groups of fMr.

## Model Theory

Lecture Series IX: Invariant random subgroups

Speaker: Simon Thomas, Rutgers University

Dates:

(1) Monday, November 5, 11 a.m.

(2) Tuesday, November 6, 11 a.m. also discussion session at 3 p.m.

(3) Wednesday, November 7, 11 a.m.

Abstract:

Let G be a countably infinite group and let SubG be the compact space of subgroups $H \le G$. Then an invariant random subgroup (IRS) of G is a probability measure on SubG which is invariant under the conjugation action of G on SubG. In particular, if $\nu$ is ergodic, then $\nu$ is a notion of randomness on SubG with a 0-1 law for every group-theoretic property. This series of lectures will be an introduction to the theory of invariant random subgroups. I will mainly focus on the case when G is a locally finite group; in this case, there are attractive connections with representation theory and the theory of finite permutation groups. In particular, I will discuss Vershik's classication of the ergodic IRSs of the group Fin(N) of finite permutations of N. In addition, I will point out some of the many open questions in this relatively new area of group theory.