$z$-classes of isometries of the hyperbolic space

Authors:
Krishnendu Gongopadhyay and Ravi S. Kulkarni

Journal:
Conform. Geom. Dyn. **13** (2009), 91-109

MSC (2000):
Primary 51M10; Secondary 51F25

DOI:
https://doi.org/10.1090/S1088-4173-09-00190-8

Published electronically:
March 26, 2009

MathSciNet review:
2491719

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 âdynamical typesâ 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, 49â87 and Conformal geometry (Bonn, 1985/1986), 41â64, Aspects Math., E12, Vieweg, Braunschweig, 1988, for previous attempts. Ahlfors (Differential Geometry and Complex Analysis (Springer, 1985), 65â73) suggested the use of Clifford algebras to deal with higher dimensional hyperbolic geometry; cf. Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 15â27, Quasiconformal Mappings and Analysis (Springer, 1998), 109â139, Complex Variables Theory Appl. 15 (1990), 125â133, and Adv. Math. 101 (1993), 87â113. 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 âIwasawa decompositionâ 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.

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Additional Information

**Krishnendu Gongopadhyay**

Affiliation:
Indian Institute of Technology (Bombay), Powai, Mumbai 400076, India

Address at time of publication:
School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India

MR Author ID:
866190

Email:
krishnendug@gmail.com

**Ravi S. Kulkarni**

Affiliation:
Indian Institute of Technology (Bombay), Powai, Mumbai 400076, India

Email:
punekulk@yahoo.com

Keywords:
Hyperbolic space,
isometry group,
dynamical types,
$z$-classes

Received by editor(s):
August 8, 2007

Published electronically:
March 26, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.