Uniform congruence counting for Schottky semigroups in ${\mathrm{SL}}_2(\mathbf{Z})$

Abstract

Let $\Gamma$ be a Schottky semigroup in ${\mathrm{SL}}_2(\mathbf{Z})$, and for $q\in \mathbf{N}$, let $Γ(q):=\{γ\in Γ: γ= e \text{ (mod $q$)}\}$ be its congruence subsemigroup of level $q$. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls $B_R$ in $M_2(\mathbf{R})$ of radius $R$: for all $q$ with no small prime factors, $(\Gamma (q)\cap B_R)=c_{\Gamma }\frac{R^{2\delta }}{({\mathrm{SL}}_2(\mathbf{Z}/q\mathbf{Z}))}+O(q^C{R}^{2\delta -\epsilon })$ as $R\to \infty$ for some $c_{\Gamma }>0,C>0,\epsilon >0$ which are independent of $q$. Our technique also applies to give a similar counting result for the continued fractions semigroup of ${\mathrm{SL}}_2(\mathbf{Z})$, which arises in the study of Zaremba's conjecture on continued fractions.

40 pages, with an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee (7 pages). This article supersedes arXiv:1412.4284 and arXiv:1507.07993. This is the final version accepted to Crelle's journal. The proof of Theorem 41 has been corrected