A product theorem in free groups

If A is a nite subset of a free group with at least two noncommuting elements, thenjA A Aj jAj 2 (logjAj)O(1) . More generally, the same conclusion holds in an arbitrary virtually free group, unless A generates a virtually cyclic subgroup. The central part of the proof of this result is carried on by estimating the number of collisions in multiple products A1 ::: Ak. We include a few simple observations showing that in this \statistical" context

[1]  H. A. Helfgott,et al.  Growth in groups: ideas and perspectives , 2013, 1303.0239.

[2]  Alexander Lubotzky,et al.  Expander graphs in pure and applied mathematics , 2011, 1105.2389.

[3]  E. Kowalski.  Sieve in expansion , 2010, 1012.2793.

[4]  H. Helfgott.  Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$ , 2008 .

[5]  J. Bourgain,et al.  Uniform expansion bounds for Cayley graphs of SL2(Fp) , 2008 .

[6]  T. Tao.  Product set estimates for non-commutative groups , 2006, Comb..

[7]  G. Freiman.  Foundations of a Structural Theory of Set Addition , 2007 .

[8]  Terence Tao,et al.  Additive combinatorics , 2007, Cambridge studies in advanced mathematics.

[9]  Mei-Chu Chang.  PRODUCT THEOREMS IN SL2 AND SL3 , 2006, Journal of the Institute of Mathematics of Jussieu.

[10]  A. Myasnikov,et al.  Elementary theory of free non-abelian groups , 2006 .

[11]  Z. Sela.  Diophantine geometry over groups VI: the elementary theory of a free group , 2006 .

[12]  Avi Wigderson,et al.  2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction , 2006, STOC '06.

[13]  H. Helfgott.  Growth and generation in SL_2(Z/pZ) , 2005, math/0509024.

[14]  Avi Wigderson,et al.  Extracting randomness using few independent sources , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[15]  A. Myasnikov,et al.  Implicit function theorem over free groups , 2003, math/0312509.

[16]  Terence Tao,et al.  A sum-product estimate in finite fields, and applications , 2003, math/0301343.

[17]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .

[18]  W. T. Gowers,et al.  A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four , 1998 .

[19]  Melvyn B. Nathanson,et al.  Additive Number Theory: Inverse Problems and the Geometry of Sumsets , 1996 .

[20]  Melvyn B. Nathanson,et al.  Additive Number Theory , 1996 .

[21]  P. Jolissaint.  Rapidly decreasing functions in reduced *-algebras of groups , 1990 .

[22]  A. Razborov.  ON SYSTEMS OF EQUATIONS IN A FREE GROUP , 1985 .

[23]  G. Makanin.  EQUATIONS IN A FREE GROUP , 1983 .

[24]  J. L. Britton.  THE BURNSIDE PROBLEM AND IDENTITIES IN GROUPS (Ergebnisse der Mathematik und ihrer Grenzgebiete. Bd. 95) , 1981 .

[25]  Sergei Ivanovich Adian,et al.  The Burnside problem and identities in groups , 1979 .

[26]  U. Haagerup.  An example of a non nuclearC*-algebra, which has the metric approximation property , 1978 .

[27]  Basil Gordon.  Review: G. A. Freiman, Foundations of a structural theory of set addition , 1975 .

[28]  Arthur Steinberg,et al.  On equations in free groups. , 1971 .