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

Anthony Lee

TL;DR

This paper classifies maximal nonelementary Fuchsian subgroups of the d=2 Bianchi group $\Gamma = \operatorname{PSL}(2, \mathcal{O}_2)$ by realizing each as the norm-$1$ group of a specific $\mathbb{Z}$-order in the indefinite quaternion algebra $Q = (-2,D)_{\mathbb{Q}}$, with the circle stabilizers in $\mathbb{C}$ encoded by reduced forms and discriminants $D$. It explicitly constructs six $\mathbb{Z}$-orders $\mathcal{M}_{(i)}$ whose reduced-norm-1 groups yield all $F$-subgroups, detailing their $\mathbb{Z}$-bases and the congruence conditions on $D$ that select each order. The authors derive a general volume formula for these subgroups, including a careful 2-adic analysis of the Eichler symbols at $p=2$ and the reduced-norm indices, to produce exact covolumes. They further obtain an asymptotic for the number of primitive immersed totally geodesic surfaces of bounded area, revealing a linear growth with a computable constant, and thereby provide precise prime-geodesic-type results for $d=2$ Bianchi orbifolds under certain class group hypotheses. The work advances explicit arithmetic descriptions of totally geodesic submanifolds in Bianchi manifolds and connects quaternionic orders, circle stabilizers, and geometric invariants in a concrete, computable framework.

Abstract

Let $Γ$ denote the $d = 2$ Bianchi group $\operatorname{PSL}(2,\mathbb{Z}[\sqrt{-2}])$. We give an explicit description of all conjugacy classes of maximal nonelementary Fuchsian subgroups of $Γ$ as integral orders of certain indefinite quaternion algebras over $\mathbb{Q}$. Using this description, we also provide the covolumes corresponding to each conjugacy class. As an application, we compute the limit $\lim_{x\to\infty} \frac{Π(x)}{x}$ where $Π(x)$ counts the number of primitive totally geodesic immersed surfaces in the manifold $Γ\backslash\mathbb{H}^3$ with area less than $x$.

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

TL;DR

This paper classifies maximal nonelementary Fuchsian subgroups of the d=2 Bianchi group by realizing each as the norm- group of a specific -order in the indefinite quaternion algebra , with the circle stabilizers in encoded by reduced forms and discriminants . It explicitly constructs six -orders whose reduced-norm-1 groups yield all -subgroups, detailing their -bases and the congruence conditions on that select each order. The authors derive a general volume formula for these subgroups, including a careful 2-adic analysis of the Eichler symbols at and the reduced-norm indices, to produce exact covolumes. They further obtain an asymptotic for the number of primitive immersed totally geodesic surfaces of bounded area, revealing a linear growth with a computable constant, and thereby provide precise prime-geodesic-type results for Bianchi orbifolds under certain class group hypotheses. The work advances explicit arithmetic descriptions of totally geodesic submanifolds in Bianchi manifolds and connects quaternionic orders, circle stabilizers, and geometric invariants in a concrete, computable framework.

Abstract

Let denote the Bianchi group . We give an explicit description of all conjugacy classes of maximal nonelementary Fuchsian subgroups of as integral orders of certain indefinite quaternion algebras over . Using this description, we also provide the covolumes corresponding to each conjugacy class. As an application, we compute the limit where counts the number of primitive totally geodesic immersed surfaces in the manifold with area less than .
Paper Structure (8 sections, 7 theorems, 83 equations)

This paper contains 8 sections, 7 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Obtaining the Integral Orders
  3. The orders $\mathcal{M}_{(2)}$, $\mathcal{M}_{(5)}$, $\mathcal{M}_{(6)}$
  4. The orders $\mathcal{M}_{(3)}$, $\mathcal{M}_{(4)}$
  5. Volume Formulae
  6. Computing the Eichler symbol of $\mathcal{M}$ at $p=2$
  7. Computing the Index of the Reduced Norm Group at $p = 2$
  8. Counting Prime Geodesic Surfaces

Key Result

Theorem 2

Let $D$ be the discriminant of the equation of the circle corresponding to an $F$-subgroup. Then each conjugacy class of $F$-subgroups can be represented as the group of reduced norm 1 elements of the following $\mathbb{Z}$-orders in $Q$:

Theorems & Definitions (13)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Proposition 7: Vulakh_1991, Theorem 3
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 3 more