See also my website tointon.neocities.org.

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. One of the central themes of my research is to develop, add to, and further exploit these links.

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.

A classical way of viewing a finitely generated group geometrically, on the other hand, 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 ∈ S such that x = ys. A famous theorem of Gromov shows that a certain 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. For example, a simple random walk on G is a random sequenece x1, x2, . . . of elements of G, in which x1 is the identity and then each xn+1 is chosen uniformly at random from the neighbours of xn in C(G,S). This then has 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 (dare I write it?) the spread of a virus.

It turns out that all of these different aspects of group theory are related. For example, Gromov’s theorem has important consequences for the behaviour of random walks and percolation on groups. Moreover, in 2011, Breuillard, Green and Tao used approximate groups to give a refinement of Gromov’s theorem, which in turn can be used to refine and significantly generalise some of these consequences.