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

Abstract

Let $Γ$ 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, $ (Γ(q) \cap B_R )= c_Γ\frac{R^{2δ}}{ (\mathrm{SL}_2(\mathbf{Z}/q\mathbf{Z}))} +O(q^C R^{2δ-ε})$ as $R\to \infty$ for some $c_Γ>0, C>0, ε>0$ which are independent of $q$. Our technique also applies to give a similar counting result for the continued fractions semigroup of $\mathrm{SL}_2(\mathbf{Z})$, 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