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

TL;DR

This paper proves that nilprogressions can be approximated by proper coset nilprogressions, and applies this to confirm a conjecture on polynomial growth in groups, with implications for geometric group theory.

Contribution

It establishes a properness result for nilprogressions and verifies a conjecture on polynomial growth, extending the understanding of group growth behavior.

Findings

Nilprogressions can be approximated by proper coset nilprogressions.

Confirmed Benjamini's conjecture on polynomial growth of symmetric generating sets.

Provided tools for analyzing scaling limits of vertex-transitive graphs.

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 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 $|S^n|\le Mn^D$ 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 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.