z-CLASSES OF ISOMETRIES OF THE HYPERBOLIC SPACE

Let G be a group. Two elements x, y are said to be z-equivalent if their centralizers are conjugate in G. The class equation of G is the partition of G into conjugacy classes. Further decomposition of conjugacy classes into z-classes provides an important information about the internal structure of the group, cf. [8] for the elaboration of this theme. Let I(Hn) denote the group of isometries of the hyperbolic n-space, and let Io(H n) be the identity component of I(Hn). We show that the number of z-classes in I(Hn) is finite. We actually compute their number, cf. theorem 1.3. We interpret the finiteness of z-classes as accounting for the finiteness of “dynamical types” in I(Hn). Along the way we also parametrize conjugacy classes. We mainly use the linear model of the hyperbolic space for this purpose. This description of parametrizing conjugacy classes appears to be new, cf. [4], [9] for previous attempts. Ahlfors [1] suggested the use of Clifford algebras to deal with higher dimensional hyperbolic geometry, cf [2], [5], [13], [14]. These works may be compared to the approach suggested in this paper. In dimensions 2 and 3, by remarkable Lie-theoretic isomorphisms, Io(H ) and Io(H ) can be lifted to GLo(2, R), and GL(2, C) respectively. For orientation-reversing isometries there are some modifications of these liftings. Using these liftings, in the appendix A, we have introduced a single numerical invariant c(A), to classify the elements of I(H) and I(H), and explained the classical terminology. Using the “Iwasawa decomposition” of Io(H n), it is possible to equip Hn with a group structure. In the appendix B, we visualize the stratification of the group Hn into its conjugacy and z-classes.