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

Abstract

In this paper, the two settings we are concerned with are $\Gamma <\mathrm{SO}(n,1)$ a Zariski dense Schottky semigroup and $\Gamma <{\mathrm{SL}}_2(\mathbb{C})$ 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 ${\mathrm{SL}}_2(\mathbb{R})$ 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 [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).

arXiv comment42 pages, 1 figure; minor improvements added during publication

References in corpus (1)

• Exponential Mixing of Frame Flow for Convex Cocompact Hyperbolic Manifolds