Length and Torsion in Arithmetic Hyperbolic Orbifolds

Abstract

In this chapter, we will discuss the structure and properties of the set of closed geodesics, particularly in arithmetic hyperbolic 2- and 3-orbifolds. As discussed briefly in §5.3.4, this is closely connected to properties of loxodromic elements in Kleinian or Fuchsian groups, and in the case of arithmetic groups, the traces and eigenvalues of these loxodromic elements carry extra arithmetic data that can be used to help understand the set of geodesics in arithmetic hyperbolic 3-manifolds. We also consider torsion that arises in arithmetic Fuchsian and Kleinian groups. Although, on the face of things, this appears to have little to do with lengths, the existence of torsion and eigenvalues of loxodromic elements in arithmetic groups is closely tied to the algebra and number theory of the invariant trace field and quaternion algebra. In particular, their existence depends on the existence of embeddings into the quaternion algebra of suitable quadratic extensions of the defining field. Such embeddings were characterised in Chapter 7 and these results are refined in this chapter to consider embeddings of orders inside these quadratic extensions into orders in the quaternion algebras.