Wohlfahrt's Theorem and index formula for elementary matrix groups and $SL(2, \mathcal O)$

TL;DR

This paper extends Wohlfahrt's Theorem and computes indices of principal congruence subgroups for Bianchi groups, $SL(2, \\mathcal O)$, and elementary matrix groups over rings of integers in number fields.

Contribution

It generalizes Wohlfahrt's Theorem and provides explicit index formulas for principal congruence subgroups in these algebraic groups.

Findings

Determined indices of principal congruence subgroups for Bianchi groups and $SL(2, \\mathcal O)$

Extended Wohlfahrt's Theorem to these groups

Provided formulas for indices of elementary matrix groups

Abstract

The present article determines the indices of the principal congruence subgroups of the Bianchi groups $B_d$, $SL(2,\mathcal O)$ and elementary matrix group $E$ and extends Wohlfahrt's Theorem to $B_d$, $SL(2,\mathcal O)$ and $E$, where $\mathcal O$ is the ring of integers of some number fields.