Hypercomplex hyperbolic geometry

Abstract

The rank one symmetric spaces of non-compact type are the real, complex, quaternionic and octonionic hyperbolic spaces. Real hyperbolic geometry is widely studied complex hyperbolic geometry less so, whilst quaternionic hyperbolic geometry is still in its infancy. The purpose of this thesis is to investigate the conditions for discrete group action in quaternionic and octonionic hyperbolic 2-spaces and their geometric consequences, in the octonionic case, in terms of lower bounds on the volumes of non-compact manifolds. We will also explore the eigenvalue problem for the 3 x 3 octonionic matrices germane to the Jordan algebra model of the octonionic hyperbolic plane. In Chapters One and Two we concentrate on discreteness conditions in quaternionic hyperbolic 2-space. In Chapter One we develop a quaternionic Jørgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic 2-space generated by two elements, one of which is either loxodromic or boundary elliptic. In Chapter Two we give a generalisation of Shimizu's Lemma to groups of isometries of quaternionic hyperbolic 2-space containing a screw-parabolic element. In Chapter Three we present the Jordan algebra model of the octonionic hyperbolic plane and develop a generalisation of Shimizu's Lemma to groups of isometries of octonionic hyperbolic 2-space containing a parabolic map. We use this result to determine estimates of lower bounds on the volumes of non-compact closed octonionic 2-manifolds. In Chapter Four we construct an octonionic Jørgensen's inequality for non-elementary groups of isometries of octonionic hyperbolic 2-space generated by two elements, one of which is loxodromic. In Chapter Five we solve the real eigenvalue problem Xv = λv, for the 3 x 3 ɸ-Hermitian matrices, X, of the Jordan algebra model of the octonionic hyperbolic plane. Finally, in Chapter Six we consider the embedding of collars about real geodesies in complex hyperbohc 2-space, quaternionic hyperbolic 2-space and octonionic hyperbolic 2-space.