Approximate Subgroups of Linear Groups

By Emmanuel Breuillard, Ben Joseph Green, Terence Tao. 2010

242 citations recorded by the source corpus.

We establish various results on the structure of approximate subgroups in linear groups such as SLn(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 $${{\rm SL}_{n}({\mathbb {F}}_{q})}$$ which generates the group must be either very small or else nearly all of $${{\rm SL}_{n}({\mathbb {F}}_{q})}$$. 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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/e8f01e58c7c96e45c1f45195f60c9f850ef4b5a9

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Citations in the corpus

Expander graphs, gonality, and variation of Galois representations (2010, methodology): “Fℓ)) ⊂ G(Fℓ) (which are well-deﬁned for almost all ℓafter ﬁxing an integral model of G). Thus Gmight be of inﬁnite index in the lattice G(Q) ∩ GLm(Z) (the “thin case”, in the terminology suggested by [9]), and neither Property (T) nor automorphic methods are applicable. There has however been much r”
Nilprogressions and groups with moderate growth (2015, background): “Such a strong bound is currently only known for finite simple groups of bounded rank (see [9, 27, 42]).”
Expansion in perfect groups (2011, background): “Now the statement is known for all finite simple groups of Lie type due to a recent breakthrough by Breuillard, Green, Tao [15], and Pyber, Szabó [52].”
The structure of approximate groups (2011, result): “Various forms of the main theorem are also known in groups of Lie type of bounded dimension, as a consequence of results of many authors [5, 7, 8, 18, 19, 30, 31, 33, 43].”
Expansion in finite simple groups of Lie type (2013, methodology): “We state here the main result from [17]:”
Notes on Thin Matrix Groups (2012, background): “…call this “superstrong” approximation and it asserts that if we choose a finite symmetric (s ∈ S if s−1∈ S) generating set S of SLn(Z), then the congruence Cayley graphs (Xq , S) form an expander family as q goes to infinity (see [Hoory et al. 2006] for the definition and properties of expanders).”
On the local-global conjecture for integral Apollonian gaskets (2012, unknown): “b] 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 Szabo [PS10]. In particular, Bourgain and Varju [BV11, Theorem 1] proved the s”
On the diameter of permutation groups (2011, background): “Classical reasons aside, this will allow us to emphasise the link to [Hel08], [Hel11], [BGT11], [PS10], and [GH10], while repeating one of the main motifs: growth results from the action of a group on a set, often, as is the case here, by conjugation.”
Expander graphs in pure and applied mathematics (2011, background): “, assuming an armative answer to some generalized form of the 1-2-3 problem (See x2.5). That work gave a push to eorts in this direction by a good number of authors ([BG3],[BG4], [BGS3], [BV], [V], [BGT2], [PS2], [S7], [SGV]). The most general result for the \ane-sieve method" as it is called n”
A Spectral Gap Theorem in $SU(d)$ (2011, result): “ in this step, originates in the work of Helfgott [H]. Let us point out, however, that, in contrast to [H], and to the subsequent work on the product theorems in SLd(p) and other ﬁnite simple groups ([BGT], [PS]), the precise size of our diagonal set is 3 not important. The construction of this almo”
Growth in groups: ideas and perspectives (2013, unknown): “SLn). Full and elegant generalizations to all ﬁnite simple groups of Lie type (with bounds depending on the rank) were given by Pyber and Szabo´ [PSa] and, independently, by Breuillard, Green and Tao [BGT11]; this, of course, covered the classical groups SLn(Fq), SOn(Fq), Sp2n(Fq), with δ>0 depen”