Stable group theory and approximate subgroups
August 24, 2010
Nov. 12: some typos corrected, thanks to Elisabeth.
Dec. 27: Remark 6 to Theorem 3.4 was corrected and now reads as follows:
(Locality) Inspection of the proof will show that for all assertions except the normality of $S$, we only use $\mu$ (as as S1 ideal) on definable subsets of $X X \inv X$. To show normality $S$, we also require $X a X \inv$, where
$a \in X$ or $a \in X \inv$.
Thanks to Lou Van den Dries for bringing this issue to my attention.
An example is given showing this is necessary, at least if only right-invariance
is assumed. Other comments by Lou, Ward Henson, Laci Pyber are also included.
Lou's notes on his home page include a better treatment,
restricted to within $X X \inv X$, provided that $1 \in X$.
Updated Feb. 7 (comments to Lemma 4.5 by Francoise Point.)
Updated April 9, with various local corrections, most notably: the statements
of random 3-amalgamation now has a strengthened hypothesis and a proof; previously
there was only a sketch, in which Pierre Simon noted a gap. The applications
to the strict order property in section 6 are removed for possible repair, following
a comment by Krzysztof Krupinski. (Neither of these are needed for the main line of results.)
Updated June 1, see section 7.
Updated June 7, July 28, thanks to Anand for comments.
A new \S 7 has been added, combining
Gromov's proof with the methods of this paper to give an application to approximate subgroups of a fixed finitely generated group.
Sep 8, a simplification by Emmanuel Breuillard incorporated (end of \S 7.)