Romain Tessera

A quantitative version of Gromov's theorem

In joint work with Matt Tointon, we give an effective proof of the fact that if (G_n,S_n) is a sequence of Cayley graphs such that |S_nⁿ| << n^D|S_n| , then the sequence (G_n, (1/n)d_Sn) is relatively compact for the Gromov-Hausdorff topology and every cluster point is a connected nilpotent Lie group equipped with a left-invariant sub-Finsler metric. This enables us to show that the dimension of such a cluster point is bounded by D, and that, under the stronger bound |S_nⁿ| << n^D, the homogeneous dimension of a cluster point is bounded by D.
Our approach is roughly to use a well-known structure theorem for approximate groups due to Breuillard, Green and Tao to replace S_nⁿ with a coset nilprogression of bounded rank, and then to use results about nilprogressions from a previous paper of ours to study the ultralimits of such coset nilprogressions. As an application we sharply bound the dimension of the scaling limit of a sequence of vertex-transitive graphs of large diameter. We also recover and effectivise parts of an argument of Tao concerning the further growth of single set S satisfying the bound |Sⁿ| <= Mn^D|S|.