Polynomial growth and asymptotic dimension

TL;DR

This paper refines the relationship between polynomial growth rates of graphs and their asymptotic dimension, establishing sharper bounds and exploring implications for Riemannian manifolds and Assouad-Nagata dimension.

Contribution

It improves previous results by showing that graphs with polynomial growth less than n^{k+1} have asymptotic dimension at most k, and discusses related geometric implications.

Findings

Graphs with polynomial growth less than n^{k+1} have asymptotic dimension at most k.

Riemannian manifolds with bounded geometry and polynomial growth less than n^{k+1} have asymptotic dimension at most k.

Existence of graphs with growth less than n^{1+ε} but infinite asymptotic Assouad-Nagata dimension.

Abstract

Bonamy et al \cite{BBEGLPS} showed that graphs of polynomial growth have finite asymptotic dimension. We refine their result showing that a graph of polynomial growth strictly less than $n^{k+1}$ has asymptotic dimension at most $k$. As a corollary Riemannian manifolds of bounded geometry and polynomial growth strictly less than $n^{k+1}$ have asymptotic dimension at most $k$. We show also that there are graphs of growth $<n^{1+\epsilon}$ for any $\epsilon >0$ and infinite asymptotic Assouad-Nagata dimension.