Combinatorial and probabilistic aspects of groups

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A seminal theorem of Gromov from the early 80s states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. This has in turn had deep applications in probability on groups, such as Varopoulos's characterisation of groups on which the simple random walk is recurrent, and Duminil-Copin, Goswami, Raoufi, Severo and Yadin's recent breakthrough showing that the critical percolation probability on a group is strictly less than 1 provided that the group is not virtually cyclic. In fact, both of these results extend to vertex-transitive graphs using a result of Trofimov showing that a vertex-transitive graph of polynomial growth can be approximated by a Cayley graph in a certain sense. In 2011, Breuillard, Green and Tao gave a finitary refinement of Gromov's theorem as a consequence of their celebrated work on approximate groups. Tessera and Tointon subsequently obtained a similar refinement of Trofimov's result. Hutchcroft and Tointon have used these results to obtain an analogue of Duminil-Copin et al's result for finite vertex-transitive graphs, essentially verifying a conjecture of Benjamini from 2001. In forthcoming but already announced work, Tessera and Tointon develop the theory further and use it to prove a quantitative and finitary version of Varopoulos's result, part of which was conjectured by Benjamini and Kozma in 2002, and Easo and Hutchcroft use it to prove Schramm's notorious locality conjecture for percolation. Tessera and Tointon have brought this theory to the point where most of the quantitative bounds are provably sharp. However, there are a number of aspects of the theory that could still be sharpened, and one aim of this project is to sharpen up some of these. For instance, Tao has shown that the 'local' growth degree of a group of polynomial growth can both increase and decrease at most finitely many times, with the number of changes bounded in terms of the initial growth rate. Tessera and Tointon obtain a sharp bound on the number of times the degree can decrease, but their bound on the number of times it can increase may not be optimal. For another example, Breuillard and Tointon proved that a finite group with a polynomially large diameter has a 'large' quotient that has an abelian subgroup of bounded rank and index. Tessera and Tointon obtain the optimal bound on the rank, but their bound on the index may not be optimal. Moreover, it appears likely that additional structural assumptions about the group should lead to stronger bounds on the rank. The methodology of this research is by its very nature highly novel, since these questions involve improving upon a theory that is itself not yet fully published. Given the spectacular recent success of this theory in shedding light on old and difficult conjectures in probability theory, the potential for impact is also significant. This project falls within the EPSRC Algebra research area.