Skip to main content

Research Repository

Advanced Search

Uniform congruence counting for Schottky semigroups in SL2(𝐙)

Magee, Michael; Oh, Hee; Winter, Dale

Uniform congruence counting for Schottky semigroups in SL2(𝐙) Thumbnail


Authors

Hee Oh

Dale Winter



Abstract

Let Γ be a Schottky semigroup in SL2(Z), and for q∈N, let Γ(q):={γ∈Γ:γ=e(modq)} 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, #(Γ(q)∩BR)=cΓR2δ#(SL2(Z/qZ))+O(qCR2δ−ϵ) as R→∞ for some cΓ>0,C>0,ϵ>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.

Citation

Magee, M., Oh, H., & Winter, D. (2019). Uniform congruence counting for Schottky semigroups in SL2(𝐙). Journal für die reine und angewandte Mathematik, 2019(753), 89-135. https://doi.org/10.1515/crelle-2016-0072

Journal Article Type Article
Acceptance Date Nov 28, 2016
Online Publication Date Jan 12, 2017
Publication Date Jul 31, 2019
Deposit Date Sep 6, 2017
Publicly Available Date Jan 12, 2018
Journal Journal für die reine und angewandte Mathematik
Print ISSN 0075-4102
Electronic ISSN 1435-5345
Publisher De Gruyter
Peer Reviewed Peer Reviewed
Volume 2019
Issue 753
Pages 89-135
DOI https://doi.org/10.1515/crelle-2016-0072

Files


Accepted Journal Article (517 Kb)
PDF

Copyright Statement
The final publication is available at www.degruyter.com




You might also like



Downloadable Citations