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

Abstract

Let $\Gamma$ denote the $d=2$ Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\sqrt{-2}])$. 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 $x$.

12 pages