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11 - Applications to Growth in Groups
Matthew C. H. Tointon, University of Cambridge
Book:

Introduction to Approximate Groups

Published online:

31 October 2019

Print publication:

14 November 2019, pp 178-197
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Summary

We present various applications of Breuillard, Green and Tao’s rough classification of finite approximate groups to groups of polynomial growth. We define polynomial, exponential and intermediate growth, and show that these concepts are stable under changes of generating set and passing to subgroups of finite index. We prove Breuillard, Green and Tao’s result that if a ball of large enough radius in a Cayley graph is of size polynomial in the radius then the underlying group is virtually nilpotent. We deduce that all larger balls also have polynomial bounds on their sizes. We guide the reader in the exercises to Breuillard and Tointon’s results that a finite group of large diameter admits large virtually nilpotent and virtually abelian quotients. We also prove the same authors’ result that a finite simple group has diameter bounded by a small power of the size of the group. We prove an isoperimetric inequality for finite groups due to Breuillard, Green and Tao. Finally, we give a brief high-level introduction to applications of approximate groups to the construction of expanders.

Matthew C. H. Tointon
Published online:

31 October 2019

Print publication:

14 November 2019
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Approximate groups have shot to prominence in recent years, driven both by rapid progress in the field itself and by a varied and expanding range of applications. This text collects, for the first time in book form, the main concepts and techniques into a single, self-contained introduction. The author presents a number of recent developments in the field, including an exposition of his recent result classifying nilpotent approximate groups. The book also features a considerable amount of previously unpublished material, as well as numerous exercises and motivating examples. It closes with a substantial chapter on applications, including an exposition of Breuillard, Green and Tao's celebrated approximate-group proof of Gromov's theorem on groups of polynomial growth. Written by an author who is at the forefront of both researching and teaching this topic, this text will be useful to advanced students and to researchers working in approximate groups and related areas.

Assouad–Nagata Dimension of Wreath Products of Groups
N. Brodskiy, J. Dydak, U. Lang
Journal:

Canadian Mathematical Bulletin / Volume 57 / Issue 2 / 14 June 2014

Published online by Cambridge University Press:

20 November 2018, pp. 245-253

Print publication:

14 June 2014
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Consider the wreath product $H\,\wr \,G$, where $H\,\ne \,1$ is finite and $G$ is finitely generated. We show that the Assouad–Nagata dimension ${{\dim}_{AN}}\left( H\,\wr \,G \right)$ of $H\,\wr \,G$ depends on the growth of $G$ as follows: if the growth of $G$ is not bounded by a linear function, then ${{\dim}_{AN}}\left( H\,\wr \,G \right)\,=\,\infty$; otherwise ${{\dim}_{AN}}\left( H\,\wr \,G \right)\,=\,{{\dim}_{AN}}\left( G \right)\,\le \,1$.