On the spectral gap for infinite index “congruence” subgroups of SL2(Z)

By Alex Gamburd. 2002

121 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8c90213bf280a1e920624bd766dd18798828a91b

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

On representations of integers in thin subgroups of SL(2,Z) (2010, methodology): “This is Theorem 1.13 in [BKS09], using Gamburd’s [Gam02] spectral gap Θ = 5/6.”
Generalization of Selberg's 3/16 Theorem and Affine Sieve (2009, methodology): “The argument in section 2 establishes that 1 ⇒ 2; the implication 2 ⇒ 1 is proved using Fell’s continuity of induction in section 7 of [11].”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “, [19], [43]), the number R(F,w00) can also be explicitly computed in principle.”
Uniform expansion bounds for Cayley graphs of SL2(Fp) (2008, background): “In [7] it is proved that if S is a set of elements in SL2(Z) such that Hausdorff dimension of the limit set1 of 〈S〉 is greater than 5/6, then G(SL2(Fp), Sp) form a family of expanders.”
Affine linear sieve, expanders, and sum-product (2010, methodology): “The proof of the expander property follows the method in [24, 54] which is based on an upper bound on the number of closed cycles combined with exploiting the large dimensionality of a nontrivial irreducible representation of SL2(Z/pZ) (the latter is due to Frobenius [22]).”
Notes on Thin Matrix Groups (2012, background): “Thanks to a number of major developments by many people [Sarnak and Xue 1991; Gamburd 2002; Helfgott 2008; Bourgain and Gamburd 2008b; Bourgain et al. 2010; Pyber and Szabó 2010; Breuillard et al. 2011; Varjú 2012], the general expansion property is now known.”
Expansion in perfect groups (2011, background): “A few years later Gamburd [27] showed that quotients of Γ < SL2(Z) are expanders if the Hausdorff dimension of the limit set of Γ is larger than 5/6.”
Almost prime Pythagorean triples in thin orbits (2010, background): “) Collecting the results in [3], [6], [5] and the extension from prime to square-free of [12], we have: 95 Kontorovich and Oh, Almost prime Pythagorean triples in thin orbits”