Scaling limits of Cayley graphs with polynomially growing balls

TL;DR

This paper investigates the geometric limits of Cayley graphs with polynomial growth, establishing bounds on the dimension of their scaling limits and applying these results to vertex-transitive graphs.

Contribution

It proves bounds on the dimension and homogeneous dimension of cluster points for Cayley graphs with polynomial growth, extending previous results and providing new tools for analysis.

Findings

Bound the dimension of cluster points by D.

Bound the homogeneous dimension of cluster points by D.

Apply results to vertex-transitive graphs with large diameter.

Abstract

Benjamini, Finucane and the first author have shown that if (G_n,S_n) is a sequence of Cayley graphs such that |S_n^n|=O(n^D|S_n|), then the sequence (G_n,d_{S_n}/n) 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. In this paper we show that the dimension of such a cluster point is bounded by D, and that, under the stronger bound |S_n^n|=O(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_n^n 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 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 |S^n| < Mn^D|S|.