Uniform congruence counting for Schottky semigroups in SL2(𝐙)
Michael J. Magee, Hee Oh, Dale Winter · Journal für die reine und angewandte Mathematik (Crelles Journal) · 2017
Abstract Let Γ be a Schottky semigroup in SL2(𝐙) {\mathrm{SL}_{2}(\mathbf{Z})} , and for q∈𝐍 {q\in\mathbf{N}} , let Γ(q):={γ∈Γ:γ=e(modq)} {\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}} be its congruence subsemigroup of levelq. 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 {B_{R}} in M2(𝐑) {M_{2}(\mathbf{R})} of radiusR: for all positive integerqwith no small prime factors, #(Γ(q)∩BR)=cΓR2δ#(SL2(𝐙/q𝐙))+O(qCR2δ-ϵ) \#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(% \mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon}) as R→∞ {R\to\infty} for some cΓ>0,C>0,ϵ>0 {c_{\Gamma}>0,C>0,\epsilon>0} which are independent ofq. Our technique also applies to give a similar counting result for the continued fractions semigroup of SL2(𝐙) {\mathrm{SL}_{2}(\mathbf{Z})} , which arises in the study of Zaremba’s conjecture on continued fractions.