Approximate Subgroups of Linear Groups

Abstract

We establish various results on the structure of approximate subgroups in linear groups such as SL n (k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SL n (F q ) which generates the group must be either very small or else nearly all of SL n (F q ). The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over an arbitrary field k and yields a classification of approximate subgroups of G(k). In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.

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“…Our focus will lie on the ideas behind the main growth results, going from SL 2 (Z/pZ) ( [Hel08], reexamined in the light of [Hel11], [BGT11], [PSa]) up to new work on the symmetric group [HS14]. We will be able to give essentially complete proofs of Theorems 1.1-1.4 below.…”

“…We will discuss them at the end of §5.4. In brief, Theorem 1.1 is now known to hold (with δ depending on the rank) for all finite simple groups of Lie type ( [PSa], [BGT11]); there are also generalizations to C [BG08b] and to solvable groups [GH].…”

“…In particular, [BGT12] does recover a proof of Gromov's theorem (close to Hrushovski's [Hru12]), but it does not seem strong enough to give useful bounds for finite groups. * * * The use of pivoting for general groups was first advocated in [Hel11], but it came to full fruition only later, thanks to [BGT11] and [PSa]: due in part to what in retrospect was a technical difficulty-namely, an intermediate result had an unnecessarily weak quantifier (see the remarks at the end of §5.2)- [Hel11] still uses a sum-product theorem at a certain point, though it does develop a more abstract setting, which we have just demonstrated here in the simplest case.…”

On the local-global conjecture for integral Apollonian gaskets

“…The result in [BG08b] was generalized in a series of papers [BG08a], [BG09], [BGS10], [Var12], [BV11] and [SGV11]. Some of these require the generalization of [Hel08] obtained independently by Breuillard, Green and Tao [BGT11] and Pyber and Szabó [PS10].…”

A nilpotent Freiman dimension lemma

“…A number of papers have appeared in the past few years attempting to prove analogues of this result in groups other than the additive group of integers, and in particular in non-commutative groups. See, for example, [1,2,3,4,6,7,9,10,11,12,14,15].…”