INVARIANT TRACE-FIELDS AND QUATERNION ALGEBRAS OF POLYHEDRAL GROUPS

Abstract

Let P be a polyhedron in H³ of finite volume such that the group Γ(P) generated by reflections in the faces of P is a discrete subgroup of Isom H³. Let Γ⁺(P) denote the subgroup of index 2 consisting entirely of orientation-preserving isometries so that Γ⁺(P) is a Kleinian group of finite covolume. Γ⁺(P) is called a polyhedral group.

As discussed in [12] and [13] for example (see §2 below), associated to a Kleinian group Γ of finite covolume is a pair (AΓ, kΓ) which is an invariant of the commensurability class of Γ; kΓ is a number field called the invariant trace-field, and AΓ is a quaternion algebra over kΓ. It has been of some interest recently (cf. [13, 16]) to identify the invariant trace-field and quaternion algebra associated to a Kleinian group Γ of finite covolume since these are closely related to the geometry and topology of H³/Γ.

In this paper we give a method for identifying these in the case of polyhedral groups avoiding trace calculations. This extends the work in [15] and [11] on arithmetic polyhedral groups. In §6 we compute the invariant trace-field and quaternion algebra of a family of polyhedral groups arising from certain triangular prisms, and in §7 we give an application of this calculation to construct closed hyperbolic 3-manifolds with ‘non-integral trace’.