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<p>In this paper, the two settings we are concerned with are Γ &lt; SO(<i>n</i>, 1) a Zariski dense Schottky semigroup and Γ &lt; SL<sub>2</sub>(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<sub>2</sub>(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 [Sar22]. 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).</p>

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Congruence counting in Schottky and continued fractions semigroups of SO(n, 1)

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • Pratyush Sarkar

摘要

In this paper, the two settings we are concerned with are Γ < SO(n, 1) a Zariski dense Schottky semigroup and Γ < SL2(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 SL2(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 [Sar22]. 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).