Nilprogressions and groups with moderate growth

TL;DR

This paper explores the structure of groups with moderate growth, showing that large-scale doubling implies uniform doubling, and applies these results to finite Cayley graphs to derive bounds on spectral properties and diameters.

Contribution

It establishes a connection between large-scale doubling in Cayley graphs and uniform doubling, providing new bounds on spectral gaps, diameters, and subgroup structures of finite groups.

Findings

Large-scale doubling implies uniform doubling at all scales.

Finite groups with moderate growth have subgroups with cyclic quotients of comparable diameter.

Universal bounds on diameters of finite simple groups independent of CFSG.

Abstract

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter. We also give bounds on the Cheeger constant, first eigenvalue of the Laplacian, and mixing time. This can be seen as a finite-group version of Gromov's theorem on groups with polynomial growth. It also improves on a result of Lackenby regarding property (tau) in towers of coverings. Another consequence is a universal upper bound on the diameter of all finite simple groups, independent of the CFSG.