Congruence subgroups of the minimal covolume arithmetic Kleinian group

We identify the normal subgroups of the orientation preserving subgroup [3, 5, 3] of the Coxeter group [3, 5, 3], with the factor group isomorphic to PSL2(Fq) with particular congruence subgroups of an arithmetic subgroup of PSL2(C) derived from a quaternion algebra over a quartic field. 1 Motivation – Hurwitz groups and HurwitzMacbeath surfaces It is a well known fact that up to isomorphy, there is the unique hyperbolic 2-orbifold of minimal volume. This orbifold is realized as the quotient X2,3,7 = ∆(2, 3, 7)\H of the upper half plane H by the triangle group ∆(2, 3, 7) of signature (2, 3, 7). This discrete subgroup of Isom(H) ∼= PSL2(R) is defined by its presentation ∆(2, 3, 7) = 〈α, β, γ | α = β = γ = αβγ = 1〉, and this presentation determines ∆(2, 3, 7) uniquely up to conjugation in PSL2(R). The particular importance of X2,3,7 and ∆(2, 3, 7) is based on the famous Theorem of Hurwitz. It states that the compact Riemann surfaces X with maximal automorphism groups, also called Hurwitz surfaces, i.e. those Riemann surfaces X of genus g ≥ 2 with |Aut(X)| = 84(g − 1), are exactly finite Galois coverings of X2,3,7. In other words, the Hurwitz surfaces are realized as quotients X = Γ\H where Γ is a finite index and torsion free normal subgroup of ∆(2, 3, 7). A finite group G is called a Hurwitz group if G is isomorphic to the automorphism group of a Hurwitz surface. Historically, the very first example of a Hurwitz group was the simple group PSL2(F7), which is realized as the