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 $\Gamma < \operatorname{SO}(n, 1)$ a Zariski dense Schottky semigroup and $\Gamma < \operatorname{SL}_2(\mathbb C)$ 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 [arXiv:1601.03705] in $\operatorname{SL}_2(\mathbb R)$ to higher dimensions. Superficially, the proof requires two separate strategies: the expander machinery of Golsefidy-Varj\'u, 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 [arXiv:2006.07787]. Secondly, we need to adapt Stoyanov's version of Dolgopyat's method to circumvent some technical issues while the main difficulty is to prove the key inputs: the local non-integrability condition (LNIC) and the non-concentration property (NCP).