C5

Thoughts on the Burnside problem

Let $\Gamma$ be a group and let $r_n(\Gamma)$ denote the

number of isomorphism classes of irreducible $n$-dimensional complex

characters of $\Gamma$. Representation growth is the study of the

behaviour of the numbers $r_n(\Gamma)$. I will give a brief overview of

representation growth.

We say $\Gamma$ has polynomial representation growth if $r_n(\Gamma)$ is

bounded by a polynomial in $n$. I will discuss a question posed by

Brent Everitt: can a group with polynomial representation growth have

the alternating group $A_n$ as a quotient for infinitely many $n$?

In the 1930's E. Artin conjectured that a form over a p-adic field of degree d has a non-trivial zero whenever n>d^2. In this talk we will discuss this relatively old conjecture, focusing on recent developments concerning quartic and quintic forms.

I will describe how a sieve method can be used to establish the Hasse principle for the variety

$$f(t)=N(x_1,\ldots,x_k),$$

where $f$ is an irreducible cubic and $N$ is a norm form for a number field satisfying certain hypotheses.

Given a residually finite group, we analyse a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called residual finiteness growth) attempts to measure how ``efficient'' of a group is at being residually finite. We review known results about this growth, such as the existence of a Gromov-like theorem in a particular case, and explain how it naturally leads to the study of a second related growth (called intersection growth). Intersection growth measures asymptotic behaviour of the index of the intersection of all subgroups of a group that have index at most n. In this talk I will introduce these growths and give an overview of some cases and properties.

This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.