# Groups St Andrews 2017 in Birmingham

## Speakers: Talks & Abstracts

### Principal Speakers

#### Michael Aschbacher: *Finite simple groups and fusion systems*

**Abstract:** The goal of these talks is to give some insight into a program to, first, classify a large subclass of the class of simple 2-fusion systems, and then, second, to use the result on fusion systems to simplify the proof of the theorem classifying the finite simple groups. We will begin with an introduction to the theory of fusion systems, and then move on to an overview of the proof of the theorem classifying the finite simple groups of component type. We then discuss how to translate that proof into the category of 2-fusion systems, and the advantages that accrue from the change in category. Finally we describe how the result on fusion systems can be used to derive a corresponding theorem on simple groups.

#### Pierre-Emmanuel Caprace: *Locally compact groups beyond Lie theory*

**Abstract:** The theory of locally compact groups was initiated at the turn of the 20th century under the impetus of Hilbert's fifth problem. After reviewing the landmarks of its history, we will discuss some key examples, and present an overview of the recent progress made in investigating the local structure of the non-discrete simple locally compact groups that are not Lie groups. We will also describe how non-discrete locally compact groups are relevant to the study of abstract groups, all of whose proper quotients are finite.

#### Radha Kessar: *On characters and p-blocks of finite simple groups*

**Abstract:** To each prime number p and to each finite group G is associated a natural partitioning of the set of complex irreducible characters of G called the p-block partition. The aim of these lectures is two-fold: firstly, to give an introduction to p-block theory, and secondly, to describe - as far as we know the answer - the p-block partitions of finite simple groups.

#### Gunter Malle: *Local-global conjectures*

**Abstract:** The ordinary and modular representation theory of finite groups abounds with open conjectures that relate the representation theory of a group to that of its local subgroups. We will present some of these conjectures as well as recent substantial progress which was obtained by way of reduction to the situation of quasi-simple groups and application of results relying on Lusztig's theory of characters of finite reductive groups.

The first part of this course, providing basics of block theory and of representation theory of finite groups of Lie type will be joint with the one by Radha Kessar.

### One Hour Speakers

#### Tim Burness: *Simple groups, generation and probabilistic methods*

**Abstract:** It is well known that every finite simple group can be generated by two elements and this leads naturally to a wide range of problems that have been the focus of intensive research in recent years, such as random generation, (2,3)-generation and so on. In this talk I will discuss some recent progress on similar problems for subgroups of simple groups, with applications to primitive permutation groups and the study of subgroup growth (this is joint work with Liebeck and Shalev). I will also recall the notion of the spread of a finite group and I will explain how probabilistic methods (based on fixed point ratio estimates for simple groups) have been used to shed light on a far reaching conjecture of Breuer, Guralnick and Kantor. Time permitting, I will finish by mentioning some related problems on the generating graph of a finite group.

#### Vincent Guirardel: *Boundaries for Out(F*_{n})

_{n})

**Abstract:** We will discuss boundaries for Out(*F _{n}*) and related groups occurring in different contexts and a view towards applications.

#### Harald Helfgott: *The diameter of the symmetric group: ideas and tools*

**Abstract:** Given a finite group *G* and a set *A* of generators, the diameter diam(Γ(*G*,*A*)) of the Cayley graph Γ(*G*,*A*) is the smallest ℓ such that every element of *G* can be expressed as a word of length at most ℓ in *A* ∪ *A*^{-1}. We are concerned with bounding diam(*G*) := max_{A} diam(Γ(*G*,*A*)).

It has long been conjectured that the diameter of the symmetric group of degree *n* is polynomially bounded in *n*. In 2011, Helfgott and Seress gave a quasipolynomial bound (exp((log *n*)^{(4+ε)})). We will discuss a recent, much simplified version of the proof, emphasising the links in commons with previous work on growth in linear algebraic groups.

#### Andrei Jaikin-Zapirain: *On ℓ*^{2}-Betti numbers and their analogues in positive characteristic

^{2}-Betti numbers and their analogues in positive characteristic

**Abstract:** Let *G* be a group, *K* a field and *A* a *n* by *m* matrix over the group ring *K*[*G*]. Let *G* = *G*_{1} > *G*_{2} > *G*_{3} > ... be a chain of normal subgroups of *G* of finite index with trivial intersection. The multiplication on the right side by *A* induces linear maps

_{i}:

*K*[

*G*/

*G*

_{i}]

^{n}→

*K*[

*G*/

*G*

_{i}]

^{m}

*v*

_{1},...,

*v*

_{n}) → (

*v*

_{1},...,

*v*

_{n})

*A*.

We are interested in properties of the sequence {dim_{K} ker φ_{i} / |*G*:*G*_{i}|}. In particular, we would like to answer the following questions.

- Is there the limit lim
_{i → ∞}dim_{K}ker φ_{i}/ |*G*:*G*_{i}|? - If the limit exists, how does it depend on the chain {
*G*_{i}}? - What is the range of possible values for lim
_{i → ∞}dim_{K}ker φ_{i}/ |*G*:*G*_{i}| for a given group*G*?

*G*if

*K*is a number field, less known if

*K*is an arbitrary field of characteristic 0 and almost unknown if

*K*is a field of positive characteristic.

In my talk I will give several motivations to consider these questions, describe the known results and present recent advances in the case where *K* has characteristic 0.

#### Donna Testerman: *Representations and subgroup structure of simple algebraic groups*

**Abstract:** Building on the fundamental work of Dynkin for the complex semisimple Lie algebras, numerous mathematicians have studied the restrictions of irreducible representations of simple algebraic groups to closed subgroups. We will give an overview of the work carried out since 1985, starting with Seitz's major contribution in the late '80's. In particular, this work illustrates the connection between the study of such restrictions and the determination of the maximal closed connected subgroups of the classical type algebraic groups.

We describe the classification of the irreducible actions of all maximal positive-dimensional closed subgroups of simple algebraic groups and highlight some interesting branching rules for non irreducible actions. This includes work of Burness, Ford, Ghandour, Marion and Cavallin.

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