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The free nilpotent group

In a multiplicative group , the commutator of two group elements is defined as (other conventions are also in use, though they are largely equivalent for the purposes of this discussion). A group is...

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Some notes on group extensions

In mathematics, one frequently starts with some space and wishes to extend it to a larger space . Generally speaking, there are two ways in which one can extend a space : By embedding into a space...

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A proof of Gromov’s theorem

A celebrated theorem of Gromov reads: Theorem 1 Every finitely generated group of polynomial growth is virtually nilpotent. The original proof of Gromov’s theorem was quite non-elementary, using an...

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254A, Notes 0: Hilbert’s fifth problem and related topics

This fall (starting Monday, September 26), I will be teaching a graduate topics course which I have entitled “Hilbert’s fifth problem and related topics.” The course is going to focus on three related...

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The Brunn-Minkowski inequality for nilpotent groups

One of the fundamental inequalities in convex geometry is the Brunn-Minkowski inequality, which asserts that if are two non-empty bounded open subsets of , then where is the sumset of and , and...

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The structure of approximate groups

Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our paper “The structure of approximate groups“, submitted to Pub. IHES. We had announced the main results of this paper in...

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254A, Notes 8: The microstructure of approximate groups

A common theme in mathematical analysis (particularly in analysis of a “geometric” or “statistical” flavour) is the interplay between “macroscopic” and “microscopic” scales. These terms are somewhat...

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254A, Notes 9: Applications of the structural theory of approximate groups

In the last set of notes, we obtained the following structural theorem concerning approximate groups: Theorem 1 Let be a finite -approximate group. Then there exists a coset nilprogression of rank and...

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A nilpotent Freiman dimension lemma

Emmanuel Breuillard, Ben Green and I have just uploaded to the arXiv the short paper “A nilpotent Freiman dimension lemma“, submitted to the special volume of the European Journal of Combinatorics in...

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254A, addendum: Some notes on nilprogressions

This is an addendum to last quarter’s course notes on Hilbert’s fifth problem, which I am in the process of reviewing in order to transcribe them into a book (as was done similarly for several other...

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