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

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

2012-03-26

arxiv.org/abs/1203.5624v1

Mathematics

Group Theory

Scientific paper

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs with bounded degree 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 proper length 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 $|X_n| = o(diam(X_n)^\delta)$ for $\delta > 1$ sufficiently small, we are able to prove, this time by elementary means, that $(X_n)$ converges to a circle.

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