Growth and generation in SL_2(Z/pZ)

By Harald Andrés Helfgott. 2005

253 citations recorded by the source corpus.

We show that every subset of SL_2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL_2(Z/pZ), every element of SL_2(Z/pZ) can be expressed as a product of at most O((log p)^c) elements of the union of A and A^{-1}, where c and the implied constant are absolute.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/522e6126d5e1017d9ee86a94ffd585814e1a5614

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Citations in the corpus

Expansion in $SL_d(O_K/I)$, $I$ square-free (2010, methodology): “They use an argument similar to Helfgott’s [18] to reduce it to a so-called sum-product”
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]).”
Random walks in compact groups (2012, background): “We discuss the problem in two different but very related settings: profinite groups, and compact Lie groups.”
Expansion in finite simple groups of Lie type (2013, background): “The product theorem for general finite simple groups (ii) of Lie type was recently established by Pyber and Szabó [PS] (building upon the earlier work of Helfgott [Hel1, Hel2] that treated the cases SL2(Fp),SL3(Fp)); in Section 5 we give an alternate derivation of this theorem using the closely rel”
Affine linear sieve, expanders, and sum-product (2010, methodology): “2 is proven in the case that q is restricted to be prime, using Helfgott’s result [30], which in turn builds crucially on sum-product theorem in Z/pZ [9, 10].”
Approximate Subgroups of Linear Groups (2010, methodology): “We remark that a key ingredient in Helfgott’s work was the sum-product theorem in Fp [10].”
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 perfect groups (2011, background): “This statement was first proved by Helfgott for SL2(Z/pZ) [35] and for SL3(Z/pZ) [36], and later it was extended by Dinai [22] for SL2(Fq) for an arbitrary finite field Fq.”
Stable group theory and approximate subgroups (2009, result): “For S = SL2(Fq) and SL3(Fp), Theorem 1.3 follows from results Helfgott [17], [18] and Dinai; for G = SL2(C) and G = SL3(Z), Theorem 1.3 follows from [5] and [9].”