$z$-classes of isometries of the hyperbolic space

Abstract:

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 important information about the internal structure of the group; cf. J. Ramanujan Math. Soc. 22 (2007), 35-56, for the elaboration of this theme.

Let $I(\mathbb {H}^n)$ denote the group of isometries of the hyperbolic $n$-space, and let $I_o(\mathbb {H}^n)$ be the identity component of $I(\mathbb {H}^n)$. We show that the number of $z$-classes in $I(\mathbb {H}^n)$ 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 $I(\mathbb {H}^n)$. 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. Academic Press, New York, 1974, 49–87 and Conformal geometry (Bonn, 1985/1986), 41–64, Aspects Math., E12, Vieweg, Braunschweig, 1988, for previous attempts. Ahlfors (Differential Geometry and Complex Analysis (Springer, 1985), 65–73) suggested the use of Clifford algebras to deal with higher dimensional hyperbolic geometry; cf. Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 15–27, Quasiconformal Mappings and Analysis (Springer, 1998), 109–139, Complex Variables Theory Appl. 15 (1990), 125–133, and Adv. Math. 101 (1993), 87–113. These works may be compared to the approach suggested in this paper.

In dimensions $2$ and $3$, by remarkable Lie-theoretic isomorphisms, $I_o(\mathbb {H}^2)$ and $I_o(\mathbb {H}^3)$ can be lifted to $GL_o(2, \mathbb {R})$, and $GL(2, \mathbb {C})$ 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 $I(\mathbb {H}^2)$ and $I(\mathbb {H}^3)$, and explained the classical terminology.

Using the “Iwasawa decomposition” of $I_o(\mathbb {H}^n)$, it is possible to equip $\mathbb {H}^n$ with a group structure. In the appendix B, we visualize the stratification of the group $\mathbb {H}^n$ into its conjugacy and $z$-classes.