# Growth in Groups and Graphs

## A UWA research collaboration award is funding visits to Perth by UWA alumnus Nick Gill and Harald Helfgott, both from the University of Bristol, where Harald is an Advanced Research Fellow and Nick is a postdoc.

Nick will be giving a series of six seminars, three of them in early May 2010 and three later in the year, with the aim of giving us the necessary background and introduction to research in this area. Harald, who is a noted leader in this field, will be visiting in September to carry on where Nick has left off.

The first part of Nick’s visit is from 1st to 14 May and his talks are scheduled as follows:

- Tuesday 4 May, 12 noon, MLR1, I: Sum-Product
- Friday 7 May, 11am, MLR3, II: Growth in Groups of Lie Type
- TBA, III: Escape

Nick has prepared a page of relevant literature for his talks, and the abstracts follow:

### I: SUM-PRODUCT

We introduce the idea of growth in groups, before focussing on the abelian setting. We take a first look at the sum-product principle, with a brief foray into the connection between sum-product results and incidence theorems.

We then focus on Helfgott’s restatement of the sum-product principle in terms of groups acting on groups.

### II: GROWTH IN GROUPS OF LIE TYPE

Since Helfgott first proved that “generating sets grow” in SL_2(p) and SL_3(p), our understanding of how to prove such results has developed a great deal. It is now possible to prove that generating sets grow in any finite group of Lie type; what is more the most recent proofs are very direct – they have no recourse to the incidence theorems of Helfgott’s original approach.

We give an overview of this new approach, which has come to be known as a “pivotting argument”. There are five parts to this approach, and we outline how these fit together.

### III: ESCAPE

The principle of “escape from subvarieties” is the first step in proving growth in groups of Lie type. We give a proof of this result, and its most important application (for us) – the construction of regular semisimple elements.

We then examine other related ideas from algebraic geometry, in particular the idea of non-singularity.

Nick is writing notes for each of his lectures, and they can be found here:

http://www.maths.bris.ac.uk/~manpg/australia.html