Congruence counting in Schottky and continued fractions semigroups of $\operatorname{SO}(n, 1)$

Abstract

In this paper, the two settings we are concerned with are Γ < SO(n, 1) a Zariski dense Schottky semigroup and Γ < SL 2 (C) a Zariski dense continued fractions semigroup. In both settings, we prove a uniform asymptotic counting formula for the associated congruence subsemigroups, generalizing the work of Magee-Oh-Winter [MOW19] in SL 2 (R) to higher dimensions. Superficially, the proof requires two separate strategies: the expander machinery of Golsefidy-Varjú, based on the work of Bourgain-Gamburd-Sarnak, and Dolgopyat's method. However, there are several challenges in higher dimensions. Firstly, using the expander machinery requires a key input: the Zariski density and full trace field property of the return trajectory subgroups, newly introduced in [Sar20]. Secondly, we need to adapt Stoyanov's version of Dolgopyat's method to circumvent some technical issues but the greatest difficulty is to prove a key input: the local non-integrability condition (LNIC).