Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Introduction
- 2 Basic Concepts
- 3 Coset Progressions and Bohr Sets
- 4 Small Doubling in Abelian Groups
- 5 Nilpotent Groups, Commutators and Nilprogressions
- 6 Nilpotent Approximate Groups
- 7 Arbitrary Approximate Groups
- 8 Residually Nilpotent Approximate Groups
- 9 Soluble Approximate Subgroups of GLn(C)
- 10 Arbitrary Approximate Subgroups of GLn(C)
- 11 Applications to Growth in Groups
- References
- Index
7 - Arbitrary Approximate Groups
Published online by Cambridge University Press: 31 October 2019
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Introduction
- 2 Basic Concepts
- 3 Coset Progressions and Bohr Sets
- 4 Small Doubling in Abelian Groups
- 5 Nilpotent Groups, Commutators and Nilprogressions
- 6 Nilpotent Approximate Groups
- 7 Arbitrary Approximate Groups
- 8 Residually Nilpotent Approximate Groups
- 9 Soluble Approximate Subgroups of GLn(C)
- 10 Arbitrary Approximate Subgroups of GLn(C)
- 11 Applications to Growth in Groups
- References
- Index
Summary
We state Breuillard, Green and Tao’s rough classification of the finite approximate subgroups of an arbitrary group. This states that a finite approximate subgroup of an arbitrary group is contained in a union of a few cosets of a finite-by-nilpotent group, the nilpotent quotient of which has bounded step. We define coset nilprogressions, and show how to deduce a more detailed version of the Breuillard–Green–Tao theorem in which the approximate subgroup is contained in a union of a few translates of a coset nilprogression of bounded rank and step.
- Type
- Chapter
- Information
- Introduction to Approximate Groups , pp. 130 - 132Publisher: Cambridge University PressPrint publication year: 2019