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

Gamburd, Alexander

doi:10.1007/bf02784530

Cited by 84 publications

(104 citation statements)

References 33 publications

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“…This lower bound is clearly in favor of the Guillopé-Zworski conjecture, at least for the class of groups considered above. One may wonder at this point if Theorem 1.3 is not empty: Gamburd [2002] has shown in (see Section 4 for details) the existence of several geometrically finite subgroups of PSL 2 ‫)ޚ(‬ with dimension δ > 3 4 . Another natural question is can we give a bound on the sequence |Re z k |?…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…[1984] shows that for geometrically finite groups, the critical exponent is also the Hausdorff dimension of the limit set, hence N and N have same dimension for their limit set. The group N is also considered in [Gamburd 2002], where he shows using a min-max argument and a suitable test function that δ( N ) can be made as close to 1 as we want, provided N is large enough (estimates are effective). An alternative way to construct similar convex cocompact subgroups of PSL 2 ‫)ޚ(‬ with δ close to 1 is given in [Bourgain and Kontorovich 2010].…”

Section: Examplesmentioning

confidence: 99%

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Lower bounds for resonances of infinite-area Riemann surfaces

Jakobson

Naud

2010

APDE

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For infinite-area, geometrically finite surfaces X = ‫ވ\‬ 2 , we prove new omega lower bounds on the local density of resonances Ᏸ(z) when z lies in a logarithmic neighborhood of the real axis. These lower bounds involve the dimension δ of the limit set of . The first bound is valid when δ > and if is an infinite-index subgroup of certain arithmetic groups. In this case we obtain a polynomial lower bound. Both results are in favor of a conjecture of Guillopé and Zworski on the existence of a fractal Weyl law for resonances.

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Section: Introduction and Resultsmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Lower bounds for resonances of infinite-area Riemann surfaces

Jakobson

Naud

2010

APDE

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“…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 SL 2 (Z/pZ) (the latter is due to Frobenius [22]). The extension of the required multiplicity bound in SL 2 (Z/qZ) is straightforward, proceeding inductively on the number of prime factors of q.…”

Section: Conjecture 14 Let G ⊂ Gl N Be Connected Simply Connected mentioning

confidence: 99%

Affine linear sieve, expanders, and sum-product

2009

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Let O be an orbit in Z n of a finitely generated subgroup of GL n (Z) whose Zariski closure Zcl( ) is suitably large (e.g. isomorphic to SL 2 ). We develop a Brun combinatorial sieve for estimating the number of points on O at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings. A fundamental role is played by the expansion property of the "congruence graphs" that we associate with O. This expansion property is established when Zcl( ) = SL 2 , using crucially sum-product theorem in Z/qZ for q square-free.

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“…In [7] it is proved that if S is a set of elements in SL 2 (Z) such that Hausdorff dimension of the limit set 1 of S is greater than 5/6, then G(SL 2 (F p ), S p ) form a family of expanders. Numerical experiments of Lafferty and Rockmore [12], [13], [14] indicated that Cayley graphs of SL 2 (F p ) are expanders with respect to projection of fixed elements of SL 2 (Z), as well as with respect to random generators.…”

Section: Introductionmentioning

confidence: 99%

“…The idea of obtaining spectral gap results by exploiting high multiplicity together with the upper bound on the number of short closed geodesics is due to Sarnak and Xue [22]; it was subsequently applied in [5] and [7]. In these works the upper bound was achieved by reduction to an appropriate diophantine problem.…”

Section: Introductionmentioning

confidence: 99%