Matthew Tointon

Scaling limits of vertex-transitive graphs of polynomial size or polynomial growth

The iconic Telstar football (see e.g. https://en.wikipedia.org/wiki/File:Telstar_Durlast.jpg) can be thought of as an approximation of a sphere by a truncated icosahedron. A natural question is: is there a better metric approximation of a sphere by a rescaled vertex-transitive graph? I don't know the answer to that, but around ten years ago Gelander proved the perhaps slightly surprising fact that a sphere cannot be approximated arbitrarily well by vertex-transitive graphs. In fact, he showed that the only compact homogeneous manifolds that can be approximated arbitrarily well by finite homogeneous metric spaces are tori. Not long after, Benjamini, Finucane & Tessera showed that if the approximating metric spaces are rescaled vertex-transitive graphs in which the number of vertices is polynomial in the diameter of the graph then the limit must be a Finsler torus. I will describe this and similar results, and explain how they can be deduced from recent work joint with Tessera with the addition of sharp bounds on the dimension of the limits.