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 $\Gamma(q):={\gamma\in \Gamma: \gamma= 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.