Algebraic Characterization of the Isometries of the Complex and Quaternionic Hyperbolic Plane

in terms of their trace and determinant are foundational in the real hyperbolic geometry. The counterpart of this characterization for isometries of H C was given by Giraud [8] and Goldman [9]. In this paper we offer algebraic characterization for the isometries of H H. The methods we follow carry over to the complex hyperbolic space, and yields an alternative characterization of the isometries of H C which is different from those of Giraud-Goldman. Two elements in a group G are said to be in the same z-class if their centralizers are conjugate in G. The z-classes of isometries of the real hyperbolic space have been described by Gongopadhyay-Kulkarni [3]. In this paper we describe the z-classes of isometries of the twodimensional complex and quaternionic hyperbolic space. In fact, we explicitly compute their number.