# 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.

## Trace fields

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.[1] 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.

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