Integral lattices and hyperbolic manifolds

In [48] Ratcliffe and Tschantz use integral Lorentzian [special characters omitted]-lattices to construct real hyperbolic manifolds. The automorphism group of such a lattice naturally acts on real hyperbolic space and for suitable subgroups the resulting quotient space is an arithmetically defined hyperbolic manifold. They use elementary methods to compute the volumes of these manifolds via the index of the subgroup. In some sense, this thesis tries to generalise this idea to produce manifolds modelled on complex and quaternionic hyperbolic space.

We address the problem of which rings best generalise the role played by [special characters omitted] for lattices in [special characters omitted] when [special characters omitted] is replaced by [special characters omitted] or [special characters omitted]. This question is tackled in some detail in Chapter 5 where we conclude that the most suitable ring in [special characters omitted] is the ring of Hurwitz integers [special characters omitted]. We study the factorisation properties and ideals of this ring, extending the results in [15].

Using a procedure of Allcock [3, 4] we find generators for the group of automorphisms of some Lorentzian R-lattices (we also follow the method of Falbel and Parker [46] to produce a set of generators in the case of a three dimensional [special characters omitted]-lattice). Having shown that finite index torsion-free subgroups will lead to finite volume complex and quaternionic hyperbolic manifolds we find precisely when the principal congruence subgroups are torsion-free. In the complex case we are then able to compute the index of these subgroups and for n = 2 we establish the volume of some of the resulting manifolds.