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Invariant Trace‐Fields and Quaternion Algebras of Polyhedral Groups
Author(s) -
MacLachlan C.,
Reid A. W.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006747
Subject(s) - mathematics , quaternion , invariant (physics) , trace (psycholinguistics) , polyhedron , discrete group , pure mathematics , quaternion algebra , group (periodic table) , algebra over a field , combinatorics , division algebra , geometry , mathematical physics , subalgebra , physics , quantum mechanics , philosophy , linguistics
Let P be a polyhedron in H 3 of finite volume such that the group Γ( P ) generated by reflections in the faces of P is a discrete subgroup of Isom H 3 . 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 ( A Γ, k Γ) which is an invariant of the commensurability class of Γ; k Γ is a number field called the invariant trace‐field, and A Γ is a quaternion algebra over k Γ. 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 H 3 /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’.