Reflections in extended Bianchi groups

Abstract

Letdbe a positive square-free number. LetOdbe the ring of integers in Q(√ −d). The groupsPSL(2, Od) are called collectively the Bianchi groups. The extended Bianchi groupBdis the maximal discrete extension ofPSL(2,Od) inPSL(2, C). The groupBdacts by linear fractional transformations on the complex plane C. LetRBdbeBdwith the generator θ,, adjoined. (RBdwill be also called an extended Bianchi group (cf. [18])). A group is said to be reflective if it contains a Coxeter subgroup (i.e. a subgroup generated by reflections) of finite index. The groupsRBdand their subgroups have been investigated in [3, 6, 13, 15, 16, 17, 18, 20, 21]. In 1892 Bianchi[3] proved thatPGL(2,Od)⋊{θ} is reflective ifd≤ 19,d╪ 14 or 17. Vinberg [18] proved that if the groupRBdis reflective, then the orders of all elements of the ideal-class group of the field Q(√ −d) should divide 4. Shaiheev [16] proved that there are only finitely many reflective extended Bianchi groups and found all of them ford≤ 30. Similar results are obtained for groupsPGL(2,Od) ⋊ {θ} which, as Shvartsman [17] showed, are reflective only whend= 1, 2, 5, 6, 10, 13, 21, providedd= 1 or 2 (mod 4).