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

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