The Arithmetic of Hyperbolic 3-Manifolds pp 111-131 | Cite as

# Invariant Trace Fields

## Abstract

The main algebraic invariants associated to a Kleinian group are its invariant trace field and invariant quaternion algebra. For a finite-covolume Kleinian group, its invariant trace field is shown in this chapter to be a number field (i.e., a finite extension of the rationals). This allows the invariants and the algebraic number-theoretic structure of such fields to be used in the study of these groups. This will be carried out in subsequent chapters. The invariant trace field is not, in general, the trace field itself but the trace field of a suitable subgroup of finite index. It is an invariant of the commensurability class of the group and that is established in this chapter. This invariance applies more generally to any finitely generated non-elementary subgroup of PSL(2, C). Likewise, the invariance, with respect to commensurability, of the associated quaternion algebra is also established. Given generators for the group, these invariants, the trace field and the quaternion algebra, can be readily computed and techniques are developed here to simplify these computations.

## Keywords

Finite Index Kleinian Group Fuchsian Group Quaternion Algebra Finite Extension## Preview

Unable to display preview. Download preview PDF.

## Further Reading

- Thurston, W. (1979). The geometry and topology of three-manifolds. Notes from Princeton University.Google Scholar
- Macbeath, A. (1983). Commensurability of cocompact three dimensional hyperbolic groups.
*Duke Math. J*, 50: 1245–1253.MathSciNetzbMATHCrossRefGoogle Scholar - Bass, H. (1980). Groups of integral representation type.
*Pacific J. Math*, 86: 15–51.MathSciNetzbMATHCrossRefGoogle Scholar - Takeuchi, K. (1975). A characterization of arithmetic Fuchsian groups.
*J. Math. Soc. Japan*, 27: 600–612.MathSciNetzbMATHCrossRefGoogle Scholar - Reid, A. (1987).
*Arithmetic Kleinian groups and their Fuchsian subgroups*. PhD thesis, Aberdeen University.Google Scholar - Reid, A. (1990). A note on trace fields of Kleinian groups.
*Bull. London Math. Soc*, 22: 349–352.MathSciNetzbMATHCrossRefGoogle Scholar - Macbeath, A. (1983). Commensurability of cocompact three dimensional hyperbolic groups.
*Duke Math. J*, 50: 1245–1253.MathSciNetzbMATHCrossRefGoogle Scholar - Vinberg, E. (1971). Rings of definition of dense subgroups of semisimple linear groups.
*Math. USSR Izvestija*, 5: 45–55.MathSciNetCrossRefGoogle Scholar - Reid, A. (1992). Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds.
*Duke Math. J*, 65: 215–228.MathSciNetzbMATHCrossRefGoogle Scholar - Helling, H., Mennicke, J., and Vinberg, E. (1995). On some generalized triangle groups and three-dimensional orbifolds.
*Trans. Moscow Math. Soc*, 56: 1–21.MathSciNetGoogle Scholar - Hilden, H., Lozano, M., and Montesinos-Amilibia, J. (1992c). A characterisation of arithmetic subgroups of SL(2, R) and SL(2,
*C). Math. Nach*, 159: 245–270.MathSciNetzbMATHCrossRefGoogle Scholar - Takeuchi, K. (1977b). Commensurability classes of arithmetic triangle groups.
*J. Fac. Sci. Univ. Tokyo*, 24: 201–222.zbMATHGoogle Scholar - Gehring, F. and Martin, G. (1989). Stability and extremality in Jorgensen’s inequality.
*Complex Variables*, 12: 277–282.MathSciNetzbMATHCrossRefGoogle Scholar