Congruence subgroups of the minimal covolume arithmetic Kleinian group

Abstract

We consider the problem of finding the normal subgroups of the orientation preserving subgroup Δ+ of the [3,5,3]-Coxeter group with the factor group isomorphic to $\operatorname{\mathrm{PSL}}_{2}(\mathbb {F}_{q})$ . We identify all such groups with particular congruence subgroups of an arithmetic subgroup of PSL 2(ℂ) derived from a quaternion algebra over a quartic field. The result can be interpreted as a generalization of the Macbeath’s result on the classification of finite linear groups as Hurwitz groups to 3-dimensional hyperbolic space.