Arithmetic hyperbolic 3-manifold
In mathematics, an arithmetic hyperbolic 3-manifold is a hyperbolic 3-manifold whose fundamental group is an arithmetic group as a subgroup of PGL(2,C). The one of smallest volume is the Weeks manifold, and the one of next smallest volume is the Meyerhoff manifold.
The trace field of a Kleinian group Γ is the field generated by the traces of representatives of its elements in SL(2, C) and it is denoted by tr Γ. The trace field of a finite covolume Kleinian group is an algebraic number field, a finite extension of the rational numbers, which is not totally real.
The invariant trace field of a Kleinian group Γ is the trace field of the Kleinian group Γ(2) generated by squares of elements of Γ.
The quaternion algebra of a Kleinian group Γ is the subring of M(2, C) generated by the trace field and the elements of Γ, and is a 4-dimensional simple algebra over the trace field if Γ is not elementary. The invariant quaternion algebra of Γ is the quaternion algebra of Γ(2). The quaternion algebra may be split, in other words a matrix algebra; this happens whenever Γ is non-elementary and has a parabolic element, in particular if it is a Kleinian group of non-compact finite covolume 3-manifold.
The invariant trace field and invariant quaternion algebra depend only on the wide commensurability class of the group as a subgroup of SL(2, C): this is known not to be the case for the trace field. Indeed, the invariant trace field is the smallest field to occur among the trace fields of finite index subgroups of Γ.
A number field occurs as the invariant trace field of an arithmetic hyperbolic 3-manifold if and only if it has just one conjugate pair of complex embeddings.