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

Pratyush Sarkar

TL;DR

The paper develops a uniform counting theory for congruence subsemigroups in two hyperbolic-group settings: a Zariski-dense Schottky semigroup in $SO(n,1)$ and a Zariski-dense continued fractions semigroup in $SL_2(\mathbb{C})$. It combines Dolgopyat-type oscillatory methods with Golsefidy–Varjú expander machinery to achieve uniform spectral bounds for congruence transfer operators, tackling high-dimensional challenges via Zariski-density of return trajectory subgroups, full trace fields, LNIC, and NCP. The core contribution is a uniform asymptotic formula for counts in level $q$ congruence subsemigroups, derived by translating spectral bounds into a congruence renewal framework and then to asymptotic counts. This advances Diophantine counting questions in higher dimensions and connects Zaremba-type problems to transfer-operator methods in non-abelian, higher-rank settings, with potential applications to uniform distribution problems in hyperbolic geometry and continued fractions theory.

Abstract

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

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

TL;DR

The paper develops a uniform counting theory for congruence subsemigroups in two hyperbolic-group settings: a Zariski-dense Schottky semigroup in and a Zariski-dense continued fractions semigroup in . It combines Dolgopyat-type oscillatory methods with Golsefidy–Varjú expander machinery to achieve uniform spectral bounds for congruence transfer operators, tackling high-dimensional challenges via Zariski-density of return trajectory subgroups, full trace fields, LNIC, and NCP. The core contribution is a uniform asymptotic formula for counts in level congruence subsemigroups, derived by translating spectral bounds into a congruence renewal framework and then to asymptotic counts. This advances Diophantine counting questions in higher dimensions and connects Zaremba-type problems to transfer-operator methods in non-abelian, higher-rank settings, with potential applications to uniform distribution problems in hyperbolic geometry and continued fractions theory.

Abstract

In this paper, the two settings we are concerned with are a Zariski dense Schottky semigroup and 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 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).

Paper Structure

This paper contains 28 sections, 52 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the proof of \ref{['thm:MainTheorem']}
  3. Organization of the paper
  4. Preliminaries
  5. Schottky semigroup setting
  6. Continued fractions semigroup setting
  7. Limit set
  8. Coding the dynamical system
  9. The dynamical system
  10. Symbolic dynamics
  11. Thermodynamics
  12. Congruence transfer operators and their uniform spectral bounds
  13. Transfer operators
  14. Uniform spectral bounds with holonomy
  15. Reduction of Property \ref{['itm:CongruenceOperatorSpectralBoundsSmall|b|']} of \ref{['thm:CongruenceOperatorSpectralBounds']}
  16. ...and 13 more sections

Key Result

Theorem 1.1

There exist $\epsilon \in (0, \delta_\Gamma)$, $C > 0$, and a nonzero $q_0 \in \mathcal{O}$ such that for all $F \in L^\star(\mathbb H^n \cup \mathbb R^{n - 1}, \mathbb R)$ and $\gamma_0 \in \Gamma$, there exists $C_0 > 0$ such that for all $x \in \tilde{\Gamma}$, square-free $q \in \mathcal{O}$ co

Figures (1)

  • Figure 1: An illustration of an example of a set of mutually disjoint trimmed disks $\{D_a^\epsilon: a \in \mathscr{A}\}$ contained in the interior of the trimmed disk $D^\epsilon$ with $\epsilon = 0.05$. The disks are indicated by their boundaries.

Theorems & Definitions (100)

  • Theorem 1.1
  • Corollary 1.2
  • Remark
  • Remark
  • Conjecture 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Definition 2.1: Schottky semigroup
  • Remark
  • Remark
  • ...and 90 more