Results 1 
3 of
3
Product set estimates for noncommutative groups
 Combinatorica
"... Abstract. We develop the PlünneckeRuzsa and BalogSzemerédiGowers theory of sum set estimates in the noncommutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freimantype inverse theorem for a special class of 2step nilpotent groups ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
Abstract. We develop the PlünneckeRuzsa and BalogSzemerédiGowers theory of sum set estimates in the noncommutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freimantype inverse theorem for a special class of 2step nilpotent groups, namely the Heisenberg groups with no 2torsion in their vertical group. 1.
Stable group theory and approximate subgroups
 J. Amer. Math. Soc
"... Abstract. We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sumproduct phenomenon. For a simple linear group G, we show that a finite subset X with XX −1 X/X  bounded is close to a finite subgroup, or else to a subset of a pro ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
Abstract. We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sumproduct phenomenon. For a simple linear group G, we show that a finite subset X with XX −1 X/X  bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Modeltheoretically we prove the independence theorem and the stabilizer theorem in a general firstorder setting. 1.
Freiman’s theorem for solvable groups
"... Abstract. Freiman’s theorem asserts, roughly speaking, if that a finite set in a torsionfree abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by Green and Ruzsa to arbitrary abelian groups, where ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. Freiman’s theorem asserts, roughly speaking, if that a finite set in a torsionfree abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by Green and Ruzsa to arbitrary abelian groups, where the controlling object is now a coset progression. We extend these results further to solvable groups of bounded derived length, in which the coset progressions are replaced by the more complicated notion of a “coset nilprogression”. As one consequence of this result, any subset of such a solvable group of small doubling is is controlled by a set whose iterated products grow polynomially, and which are contained inside a virtually nilpotent group. As another application we establish a strengthening of the MilnorWolf theorem that all solvable groups of polynomial growth are virtually nilpotent, in which only one large ball needs to be of polynomial size. This result complements recent work of BreulliardGreen, FisherKatzPeng, and Sanders. 1.