Publications mathématiques de l'IHÉS

, Volume 116, Issue 1, pp 115–221 | Cite as

The structure of approximate groups

  • Emmanuel BreuillardEmail author
  • Ben Green
  • Terence Tao


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 AA 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.


Local Group Global Group Approximate Group Escape Norm Local Homomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© IHES and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques, Bâtiment 425Université Paris Sud 11OrsayFrance
  2. 2.Centre for Mathematical SciencesCambridgeEngland
  3. 3.Department of MathematicsUCLALos AngelesUSA

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