Stable group theory and approximate subgroups

Dec. 27, 2009
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.)