Algebraic characterization of isometries of the complex and the quaternionic hyperbolic planes

Abstract

Let H F 2 denote the two dimensional hyperbolic space over F, where F is either the complex numbers ℂ or the quaternions ℍ. It is of interest to characterize algebraically the dynamical types of isometries of H F 2. For F = ℂ, such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of isometries of H ℍ 2. Our result restricts to the case F = ℂ and provides another characterization of the isometries of H ℂ 2, which is different from the characterization due to 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 provide a finite partition of the isometry group. In this paper, we describe the centralizers of isometries of H F 2 and determine the z-classes.