Maximal Fuchsian subgroups of the d=2 Bianchi group

2026/01/22 by Anthony Lee
Mathematics · #math.NT #math.GT

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Abstract

Let Γ denote the d = 2 Bianchi group PSL(2,ℤ[√(-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 ℚ. Using this description, we also provide the covolumes corresponding to each conjugacy class. As an application, we compute the limit lim_x→∞ (Π(x))/(x) where Π(x) counts the number of primitive totally geodesic immersed surfaces in Γ\backslashℍ³ with areas less than x.

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