Properness of nilprogressions and the persistence of polynomial growth of given degree

Abstract

We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of Bilu's result that a generalised arithmetic progression can be efficiently contained in a proper generalised arithmetic progression, and indeed an important ingredient in the proof is a version of Bilu's geometry-of-numbers argument carried out in a nilpotent Lie algebra. We also present some applications. We verify a conjecture of Benjamini that if $S$ is a symmetric generating set for a group such that $1\in S$ and $|S^n|\le Mn^D$ at some sufficiently large scale $n$ then $S$ exhibits polynomial growth of the same degree $D$ at all subsequent scales, in the sense that $|S^r|\ll_{M,D}r^D$ for every $r\ge n$. Our methods also provide an important ingredient in a forthcoming companion paper in which we show that if $(\Gamma_n,S_n)$ is a sequence of Cayley graphs satisfying $|S_n^n|\ll n^D$ as $n\to\infty$, and if $m_n\gg n$ as $n\to\infty$, then every Gromov-Hausdorff limit of the sequence $(\Gamma_{n},\frac{d_{S_{n}}}{m_n})$ has homogeneous dimension bounded by $D$. We also note that our arguments imply that every approximate group has a large subset with a large quotient that is Freiman isomorphic to a subset of a torsion-free nilpotent group of bounded rank and step.