Uniform congruence counting for Schottky semigroups in SL2(𝐙)

Magee, Michael; Oh, Hee; Winter, Dale

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
• Public URL: https://durham-repository.worktribe.com/output/1377110