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

TL;DR

This paper classifies all maximal nonelementary Fuchsian subgroups of the $d=2$ Bianchi group and computes related covolumes, applying these results to analyze the asymptotic growth of primitive totally geodesic surfaces.

Contribution

It provides an explicit classification of maximal Fuchsian subgroups of the Bianchi group using quaternion algebras and calculates their covolumes, with applications to counting geodesic surfaces.

Findings

Explicit classification of Fuchsian subgroups

Calculation of covolumes for each subgroup

Asymptotic limit for counting geodesic surfaces

Abstract

Let $\Gamma$ 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 $\Gamma$ 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 $\lim_{x\to\infty} \frac{\Pi(x)}{x}$ where $\Pi(x)$ counts the number of primitive totally geodesic immersed surfaces in the manifold $\Gamma\backslash\mathbb{H}^3$ with area less than $x$.