Maximal Fuchsian subgroups of the $d=2$ Bianchi group

Technical Summary: Maximal Fuchsian Subgroups of the d = 2 Bianchi Group

Problem Statement
This paper addresses the classification and geometric quantification of maximal nonelementary Fuchsian subgroups (referred to as F-subgroups) within the $d=2$ Bianchi group $\Gamma = \text{PSL}(2, \mathbb{Z}[\sqrt{-2}])$. While the classification of such subgroups for general Bianchi groups is known to be complex and dependent on the ideal class group of the underlying imaginary quadratic field, this work focuses specifically on the case $d=2$. The primary objectives are to provide an explicit description of the conjugacy classes of these subgroups as integral orders in indefinite quaternion algebras, to derive precise covolume formulae for each class, and to apply these results to compute the asymptotic growth rate of primitive totally geodesic immersed surfaces in the hyperbolic 3-orbifold $\Gamma \backslash \mathbb{H}^3$.

Methodology
The approach relies on the correspondence between F-subgroups and circles in the boundary of hyperbolic 3-space, $\partial \mathbb{H}^3 \cong \mathbb{C} \cup \{\infty\}$.

1. Geometric Classification: The author utilizes the classification results of Vul (2017) and Jung (2019) to identify that every F-subgroup stabilizes a unique circle $C$ defined by a reduced equation $A|z|^2 + Bz + \bar{B}\bar{z} + C = 0$. The discriminant $D = B\bar{B} - AC$ of these circles serves as the primary invariant.
2. Algebraic Realization: For each discriminant $D$, the paper constructs the corresponding indefinite quaternion algebra $Q = (-2, D)_{\mathbb{Q}}$. The stabilizers of the circles are realized as groups of reduced norm 1 elements ($M_1$) within specific $\mathbb{Z}$-orders $M$ of $Q$.
3. Explicit Order Construction: By analyzing the matrix representations of the stabilizers under conjugation, the author derives the explicit $\mathbb{Z}$-bases for six distinct families of orders, denoted $M(1)$ through $M(6)$, corresponding to different congruence classes of $D$ modulo 4, 8, or 16.
4. Volume Computation: The covolumes of the resulting Fuchsian groups are computed using the volume formula for arithmetic Fuchsian groups involving the reduced discriminant of the order and local indices of reduced norm groups. A significant portion of the technical work involves the 2-adic analysis of these orders, specifically calculating the Eichler symbols and the index of the reduced norm group $[ \mathbb{Z}_2^\times : \text{nrd}(M_2^\times) ]$, which requires careful examination of values modulo 8.

Key Contributions and Results

• Explicit Classification (Theorem 2): The paper provides a complete list of the $\mathbb{Z}$-orders in $Q = (-2, D)_{\mathbb{Q}}$ that correspond to the conjugacy classes of F-subgroups. These are categorized by the congruence of the discriminant $D$:
  • $M(1)$: $\mathbb{Z}[1, i, j, ij]$ for any $D$.
  • $M(2)$: $\mathbb{Z}[1, i, \frac{1+j}{2}, \frac{i+ij}{2}]$ for $D \equiv 1 \pmod 4$.
  • $M(3), M(4)$: Orders with denominators of 4, occurring for $D \equiv 2, 6 \pmod{16}$.
  • $M(5), M(6)$: Orders with denominators of 2, occurring for $D \equiv 2, 3 \pmod 4$.
• Covolume Formulae (Theorem 3): The author derives precise volume formulae for the groups $M_1/\{\pm 1\}$. The volume is given by $c \pi F(D)$, where $F(D)$ is a product involving Legendre symbols $\left(\frac{-2}{p}\right)$, and the constant $c$ depends on the specific order and the congruence class of $D$ modulo 8. Notably, the paper corrects and extends previous volume calculations by performing a rigorous 2-adic analysis that accounts for orders with basis elements having denominators of $2^2$ (orders $M(3)$ and $M(4)$), a feature not present in the $d=1$ or $d \equiv 3 \pmod 4$ cases.
• Asymptotic Growth (Theorem 5): By combining the volume formulae with a counting lemma for discriminants, the paper establishes the asymptotic behavior of the number of primitive totally geodesic immersed surfaces, $\Pi(x)$, with area less than $x$. The limit is computed as:
  $$\lim_{x \to \infty} \frac{\Pi(x)}{x} = \frac{45}{16\pi} \prod_{p>2} \left( 1 - \frac{1}{p} + \frac{1}{p + \left(\frac{-2}{p}\right)} \right)$$

Significance and Scope
The paper claims to provide the first explicit description of maximal Fuchsian subgroups for the $d=2$ Bianchi group in terms of integral quaternion orders, filling a gap left by previous works which handled $d=1$ and $d \equiv 3 \pmod 4$. The work highlights that the $d=2$ case presents unique arithmetic challenges, specifically the existence of orders with higher powers of 2 in their denominators, which necessitates a more refined 2-adic analysis than previously applied in the field.

The author notes that while the methods yield precise results for $d=2$, they may not be easily generalizable to other Bianchi groups (e.g., $d=5$) where the number of conjugacy classes for a given discriminant can be large and the congruence conditions more complex. The results contribute to a broader understanding of prime geodesic theorems for surfaces in Bianchi orbifolds, specifically under the assumption that the ideal class group of $\mathbb{Q}(\sqrt{-d})$ contains no element of order 4. The paper does not propose new experimental applications but rather consolidates the theoretical framework for counting geometric objects in these specific hyperbolic manifolds.