-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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a group. Two elements are said to be *-equivalent* if their centralizers are conjugate in . The class equation of is the partition of into conjugacy classes. Further decomposition of conjugacy classes into -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 denote the group of isometries of the hyperbolic -space, and let be the identity component of . We show that the number of -classes in is finite. We actually compute their number; cf. theorem 1.3. We interpret the finiteness of -classes as accounting for the finiteness of ``dynamical types'' in . 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 and , by remarkable Lie-theoretic isomorphisms, and can be lifted to , and 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 , to classify the elements of and , and explained the classical terminology.

Using the ``Iwasawa decomposition'' of , it is possible to equip with a group structure. In the appendix B, we visualize the stratification of the group into its conjugacy and -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

Email:
krishnendug@gmail.com

**Ravi S. Kulkarni**

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

Email:
punekulk@yahoo.com

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

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.