# Approximate subgroups of residually nilpotent groups

Authors
• 1 University of Cambridge,
Type
Published Article
Journal
Mathematische Annalen
Publisher
Springer Berlin Heidelberg
Publication Date
Jan 28, 2019
Volume
374
Issue
1
Pages
499–515
Identifiers
DOI: 10.1007/s00208-018-01795-z
PMID: 31258186
PMCID: PMC6560002
Source
PubMed Central
Keywords
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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>1$$\end{document} n > 1 such that the ball of radius n in some Cayley graph of G has cardinality bounded by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{c\log \log n}$$\end{document} n c log log n , then G is virtually \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\log n)$$\end{document} ( log n ) -step nilpotent.