Approximate subgroups of residually nilpotent groups

TL;DR

This paper demonstrates that approximate subgroups in residually nilpotent groups are closely contained within structured, finite-by-nilpotent subgroups, leading to bounds on their structure and implications for group nilpotency based on growth conditions.

Contribution

It establishes that K-approximate subgroups in residually nilpotent groups are contained in boundedly many cosets of finite-by-nilpotent subgroups with bounded step, extending structural understanding.

Findings

Approximate subgroups are contained in boundedly many cosets of finite-by-nilpotent subgroups.

Bounds depend only on the approximation parameter K and are effective.

Growth conditions imply the group is virtually (log n)-step nilpotent.

Abstract

We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there is an integer n > 1 such that the ball of radius n in some Cayley graph of G has cardinality bounded by n^(c log log n), then G is virtually (log n)-step nilpotent.