Classification of isometries of the complex and quaternionic hyperbolic spaces

Abstract

It is well-known that the dynamical classification of isometries of the real hyperbolic plane can be characterized algebraically by the traces of the matrices representing the isometries. In this thesis we generalize that result for isometries of arbitrary dimensional complex and quaternionic hyperbolic spaces using two different approaches. More generally, we classify dynamical action of matrices in SU(p, q) using the coefficients of their characteristic polynomials. After this, we concentrate on the group SU(3, 1) that acts as the isometry group of the three dimensional complex hyperbolic space. We have given a complete account of the above classification for SU(3, 1). This generalizes a theorem of Goldman for SU(2, 1). We also classify pair of loxodromic elements in SU(3, 1) that generalizes earlier work of Parker and Platis who classified pair of loxodromics in SU(2, 1).