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

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 $\Gamma\backslash\mathbb{H}^3$ with areas less than $x$.