The structure of approximate groups

By Emmanuel Breuillard, Ben Joseph Green, Terence Tao. Publications mathématiques (Bures-sur-Yvette), 2011

250 citations recorded by the source corpus.

Let K⩾1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A⋅A is covered by K left translates of A.The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups.We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups.To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert’s fifth problem in the 1950s.Applications of our main theorem include a finitary refinement of Gromov’s theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9fec61ddda317cec755795465bed19362297fa76

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Citations in the corpus

On the joint spectral radius for isometries of non-positively curved spaces and uniform growth (2018, unknown): “at the space is geodesic). Also, each of balls of radius 2 is covered by at most P balls of radius 1. So, by collecting all of those balls of radius 1, Bis covered by at most PPnballs of radius 1. In [21] balls of radius 4 instead of 2 were used to dene the bounded packing property. This is a minor ”
Growth in groups: ideas and perspectives (2013, unknown): “e conjecture was proven for ngeneral and K= Fp as [GH, Thm. 2] (joint with Pyber and Szabo´). Breuillard, Green and Tao have given to this conjecture the name of Helfgott-Lindenstrauss conjecture; in [BGT12], they proved a qualitative version with non-explicit bounds (valid even for non-algebraic gr”
Nilprogressions and groups with moderate growth (2015, background): “We also describe there an example pertaining to a non-commutative version of the Freiman-Ruzsa conjecture and showing that super-polynomial bounds are required in [10].”
Almost Finiteness and the Small Boundary Property (2018, background): “v’s theorem [17], and property (∗) (or even a weaker version of it, know as the Tempelman condition, which does not require F−1F to be (K,δ)-invariant) implies virtual nilpotence by Corollary 11.2 in [3]. This corollary shows that if G is a ﬁnitely generated group for which there is a b > 0 such ”
Approximate lattices (2016, background): “Since then, many groundbreaking results on finite approximate subgroups have been established; in particular, Breuillard, Green and Tao established in [13] their celebrated structure theorem for finite approximate groups.”