On the scaling limit of finite vertex transitive graphs with large diameter

Abstract

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\bigl({diam(X_n)^{\delta}}\bigr)$ for $\delta > 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.