The maximal discrete extension of $$SL_2(\mathcal {\scriptstyle {O}}_K)$$ S L 2 ( O K ) for an imaginary quadratic number field K

Abstract

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 $$SL_2(\mathcal {\scriptstyle {O}}_K)$$ inside $$SL_2(\mathbb {C})$$ , which uses generalized Atkin–Lehner involutions. Moreover we find a natural characterization of this group in SO(1, 3).