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Conformal Geometry and Dynamics

ISSN 1088-4173



$z$-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
Published electronically: March 26, 2009
MathSciNet review: 2491719
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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.

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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

Ravi S. Kulkarni
Affiliation: Indian Institute of Technology (Bombay), Powai, Mumbai 400076, India

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.