The maximal discrete extension of $SL_2(\mathcal{\scriptstyle{O}}_K)$ for an imaginary-quadratic number field $K$

Aloys Krieg, Joana Rodriguez, Annalena Wernz

Published 2018-11-20, updated 2018-12-21Version 2

Let $\mathcal{\scriptstyle{O}}_K$ be the ring of integers of an imaginary quadratic number field $K$. In this paper we give a new description of the maximal discrete extension of the group $SL_2(\mathcal{\scriptstyle{O}}_K)$ over the ring of integers $\mathcal{\scriptstyle{O}}_K$ of an imaginary quadratic number field $K$ inside $SL_2(\mathbb{C})$, which uses generalized Atkin-Lehner involutions. Moreover we find a natural characterization of this group in $SO(1,3)$.