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.

arXiv comment40 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

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