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

Let $${{\bf H}^{\bf 2}_{\mathbb F}}$$ denote the two dimensional hyperbolic space over $${\mathbb F}$$ , where $${\mathbb F}$$ is either the complex numbers $${\mathbb C}$$ or the quaternions $${\mathbb H}$$ . It is of interest to characterize algebraically the dynamical types of isometries of $${{\bf H}^{\bf 2}_{\mathbb F}}$$ . For $${\mathbb F=\mathbb C}$$ , such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of isometries of $${{\bf H}^{\bf 2}_{\mathbb H}}$$ . Our result restricts to the case $${\mathbb F=\mathbb C}$$ and provides another characterization of the isometries of $${{\bf H}^{\bf 2}_{\mathbb C}}$$ , 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 $${{\bf H}^{\bf 2}_{\mathbb F}}$$ and determine the z-classes. [ABSTRACT FROM AUTHOR]