Spectral Theory of Infinite-Area Hyperbolic Surfaces

Borthwick, David

doi:10.1007/978-3-319-33877-4

Cited by 105 publications

(237 citation statements)

References 0 publications

Supporting

Mentioning

233

Contrasting

Unclassified

Order By: Relevance

“…For the Schottky semigroup case, this can be proved exactly as in the proof of [1,Proposition 15.4]. For the continued fraction case it follows from (2.3) and the chain rule that for any z ∈ I,…”

Section: Set-upmentioning

confidence: 67%

See 1 more Smart Citation

Uniform congruence counting for Schottky semigroups in SL₂(𝐙)

Magee

Winter

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

Citation for published item:wgeeD wihel nd yhD ree nd interD hle @PHIUA 9niform ongruene ounting for hottky semigroups in vP@AF9D tournl f¤ ur die reine und ngewndte wthemtikF a grelles journlF F Further information on publisher's website:The nal publication is available at www.degruyter.com With an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let Γ be a Schottky semigroup in SL2(Z), and for q ∈ N, let Γ(q) := {γ ∈ Γ : γ = e (mod q)} be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls BR in M2(R) of radius R: for all positive integer q with no small prime factors,as R → ∞ for some cΓ > 0, C > 0, > 0 which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of SL2(Z), which arises in the study of Zaremba's conjecture on continued fractions.

show abstract

Section: Set-upmentioning

confidence: 67%

“…Oh was supported in part by NSF Grant DMS-1361673. 1 consider the ball of radius R with respect to the Frobenius norm:…”

Section: Introductionmentioning

confidence: 99%

Uniform congruence counting for Schottky semigroups in SL₂(𝐙)

Magee

Winter

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

show abstract

“…6 The constant √ 2 which is not present in [18] is due to the factor y 1/4 that comes from our normalization. 7 There is a typo in (2.19) of [18]. It should read P τ, s)).…”

Section: It Followsmentioning

confidence: 99%

Geometric invariants for real quadratic fields

2016

View full text Add to dashboard Cite

Abstract. To an ideal class of a real quadratic field we associate a certain surface. This surface, which is a new geometric invariant, has the usual modular closed geodesic as its boundary. Furthermore, its area is determined by the length of an associated backward continued fraction. We study the distribution properties of this surface on average over a genus. In the process we give an extension and refinement of the Katok-Sarnak formula.

show abstract

“…For a unified exposition of the spectral theory of infinite-area surfaces, the reader is referred to [7].…”

Section: Chapter 2 Preliminariesmentioning

confidence: 99%

“…(For the construction of such a function x in dimension 2, see §6.1 in [7].) The function x is called a smooth boundary defining function and the boundary ∂X will be referred to as the conformal boundary.…”

Section: Conformally Compact Manifoldsmentioning

confidence: 99%

Moments of Eisenstein Series on Convex Co-Compact Hyperbolic Manifolds

Munroe

2016

Int Math Res Notices

View full text Add to dashboard Cite

ACKNOWLEDGEMENTSI would like to thank my supervisor, Dmitry Jakobson, for the opportunity of working under his guidance and for his encouragement and support throughout all the stages of this work. I am also grateful to him for suggesting the problem addressed in this thesis.It is a pleasure for me to thank Colin Guillarmou, to whom I am especially grateful for many helpful discussions, which contributed significantly to my general understanding of the field of research to which this work belongs. I must also thank him for his exceptional hospitality during my stay atÉcole Normale Supérieure de Paris.I would also like to thank Frédéric Naud for discussions and comments that were particularly helpful in the latter phase of the research. ii ABSTRACTWe investigate the high-energy limits of the moments of Eisenstein series, when these functions are considered as real random variables over a compact subset of a convex cocompact hyperbolic manifold. In the first part, under a restriction on the Hausdorff dimension of the limit set of the fundamental group of the manifold, we prove a general formula describing all the moments of the Eisenstein series at high-energy. In particular, we show that all the odd-order moments vanish. In the second part, we study the rate of convergence of the moments in the high-energy limit. In the case of the odd-order moments, we prove that the rate of convergence is at least polynomial. As for the even-order moments, following an approach based on the work of Guillarmou and Naud [16] concerning the equidistribution of Eisenstein series, we find a polynomial error term for the fourth moment under the additional hypothesis that the manifold in question is a surface.iii

show abstract

Spectral Theory of Infinite-Area Hyperbolic Surfaces

Cited by 105 publications

References 0 publications

Uniform congruence counting for Schottky semigroups in SL₂(𝐙)

Uniform congruence counting for Schottky semigroups in SL₂(𝐙)

Geometric invariants for real quadratic fields

Moments of Eisenstein Series on Convex Co-Compact Hyperbolic Manifolds

Contact Info

Product

Resources

About

Spectral Theory of Infinite-Area Hyperbolic Surfaces

Cited by 105 publications

References 0 publications

Uniform congruence counting for Schottky semigroups in SL2(𝐙)

Uniform congruence counting for Schottky semigroups in SL2(𝐙)

Geometric invariants for real quadratic fields

Moments of Eisenstein Series on Convex Co-Compact Hyperbolic Manifolds

Contact Info

Product

Resources

About

Uniform congruence counting for Schottky semigroups in SL₂(𝐙)

Uniform congruence counting for Schottky semigroups in SL₂(𝐙)