Combinatorial, geometric and probabilistic properties of groups

The notion of a finitely generated group is at heart an algebraic one. However, in certain situations it turns out to be fruitful to view such a group from a geometric perspective, as is often the case in the field of geometric group theory, or from a combinatorial perspective, as in the field of arithmetic combinatorics. Studying probabilistic processes on groups can also be a rich source of problems and results in discrete probability. Recently, several exciting links between these different perspectives have emerged, leading to a number of breakthroughs and some beautiful results. The overriding theme of this research is to develop, add to, and further exploit these links.

A classical way of viewing a finitely generated group geometrically is to view it as a graph. If G is a group with a finite symmetric generating set S then the Cayley graph C(G,S) is the graph whose vertex set is the set of elements of G, with x and y connected by an edge if and only if there exists s E S such that x = ys. A famous theorem of Gromov shows that a certain asymptotic geometric property of this graph (polynomial growth) is equivalent to a rather strong algebraic condition on the group G (virtual nilpotence).

One can also use Cayley graphs to define various probabilistic processes on a group G, such as random walks, which have various physical interpretations, for example in the context of electric networks. Percolation on G, on the other hand, is where each edge of C(G,S) is either deleted or retained at random according to some probability distribution, and then the resulting random graph is studied. This can be interpreted in the context of the flow of water through a porous stone, or the spread of a virus. The rate of polynomial growth of a group turns out to be intimately connected to the behaviours of both random walks and percolation.

A particular example of how powerful the combinatorial perspective on groups can be is provided by objects called approximate subgroups. These are subsets of a group that are 'approximately closed' under the group operation in a certain sense. There has been considerable progress in the study of approximate subgroups over the last 15 years, and this has led to a number of remarkable applications, for example to fields as diverse as number theory, random matrix theory and theoretical computer science.

Approximate groups can also be thought of as a 'local' version of polynomial growth, and indeed a seminal result of Breuillard, Green and Tao about approximate groups can be used to show that polynomial growth on a given region of a group is enough to imply virtual nilpotence. This result has been developed by Tessera and Tointon, using 'local' versions of various group-theoretic properties, and applied to describe in detail certain fine-scale behaviours of random walks on groups, verifying and generalising two long-standing conjectures of Benjamini and Kozma. Hutchcroft and Tointon have also deployed this and additional 'local' group-theoretic notions (such as the notion of taking a quotient of an abelian group by a subset, rather than a subgroup) to analyse percolation on finite groups, verifying most cases of a famous conjecture of Benjamini.

This project will, amongst other things, seek to develop this 'local' perspective on group theory, in an effort to prove further finitary and quantitative results in a similar direction. It falls between the Algebra; Geometry & Topology; and Logic & Combinatorics EPSRC research areas.