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.
References
- ↑ "A note on Trace-Fields of Kleinian Groups". Bulletin of the London Mathematical Society 22 (4): 349–352. 1990. doi:10.1112/blms/22.4.349.
- Maclachlan, Colin; Reid, Alan W. (2003), The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics 219, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98386-8, MR 1937957