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

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 the Freiman-Bilu 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 Lie-algebra version of the geometry-of-numbers argument at the centre of that result. 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 $|S^n|\le Mn^D$atsomesufficientlylargescale$n$then$S$ exhibits polynomial growth of the same degree $D$ at all subsequent scales, in the sense that $\mathrm{\mid }{S}^r\mathrm{\mid }{\ll }_{M,D}{r}^D$ for every $r\ge n$. Our methods also provide an important ingredient in a forthcoming companion paper in which we reprove and sharpen a result about scaling limits of vertex-transitive graphs of polynomial growth due to Benjamini, Finucane and the first author. 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.