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...
View ArticleSome 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...
View ArticleA 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...
View Article254A, 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...
View ArticleThe 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...
View ArticleThe 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...
View Article254A, 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...
View Article254A, 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...
View ArticleA 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...
View Article254A, 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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