The structure of approximate groups
Breuillard, Emmanuel1 ; Green, Ben2 ; Tao, Terence3
1 Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11 91405, Orsay France
2 Centre for Mathematical Sciences Wilberforce Road, Cambridge, CB3 0WA England
3 Department of Mathematics, UCLA 405 Hilgard Ave, Los Angeles, CA, 90095 USA
Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 115-221.

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.

DOI : 10.1007/s10240-012-0043-9
Breuillard, Emmanuel 1 ; Green, Ben 2 ; Tao, Terence 3

1 Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11 91405, Orsay France
2 Centre for Mathematical Sciences Wilberforce Road, Cambridge, CB3 0WA England
3 Department of Mathematics, UCLA 405 Hilgard Ave, Los Angeles, CA, 90095 USA 
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Breuillard, Emmanuel; Green, Ben; Tao, Terence. The structure of approximate groups. Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 115-221. doi : 10.1007/s10240-012-0043-9. http://www.numdam.org/articles/10.1007/s10240-012-0043-9/

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