Congruence subgroups of the minimal covolume arithmetic Kleinian group (Q609404)

Let \(\Delta^+\) be the orientation preserving subgroup of the \([3,5,3]\)-Coxeter group. By number-theoretic methods, the author gives a list of the normal subgroups \(K\) of \(\Delta^+\) whose factor group \(\Delta^+/K\) is isomorphic to PSL\(_2(\mathbb{F}_q)\). First, the author considers the field \(F=\mathbb{Q}(\sqrt{\alpha})\), where \(\alpha=3-2\sqrt{5}\), and gives the prime ideal decomposition of the principal ideals \(p{\mathcal O}_F\) for rational primes \(p\) and the inertia degree of \(p\) at each prime ideal. Secondly, for every prime ideal \(\mathfrak{p}\) in \({\mathcal O}_F\), the author shows that the factor group of \(\Gamma_{{\mathcal M}}\) by a principal congruence subgroup \(\Gamma_{{\mathcal M}}(\mathfrak{p})\) is isomorphic to PSL\(_2({\mathcal O}_F/\mathfrak{p})\), where \(\Gamma_{{\mathcal M}}\) is defined by elements with reduced norm \(1\) of maximal order \({\mathcal M}\) in the quaternion algebra over \(F\). This result is a generalization of the study of normal subgroups of the \([2,3,7]\)-triangle group whose quotients by principal congruence subgroups are isomorphic to \(\text{PSL}_2(\mathbb{F}_q)\).