INTEGRAL LATTICES AND HYPERBOLIC REFLECTION GROUPS

The aim of this paper is to study arithmetic groups of isometries of hyperbolic spaces which are generated by hyperplane reflections. This leads to the notion of reflexive Lorenzian lattices. The main contribution of this paper is to give many new examples of such lattices in dimensions 3 and 4. These lattices give rise to maximal, pairwise non-conjugate arithmetic reflexion groups on hyperbolic 3-space, respectively 4-space. The method belongs to the arithmetic theory of quadractic forms.