## $z$-classes of isometries of the hyperbolic space

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- by Krishnendu Gongopadhyay and Ravi S. Kulkarni PDF
- Conform. Geom. Dyn.
**13**(2009), 91-109 Request permission

## 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 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.

## References

- Lars V. Ahlfors,
*MĂ¶bius transformations and Clifford numbers*, Differential geometry and complex analysis, Springer, Berlin, 1985, pp.Â 65â73. MR**780036** - Lars V. Ahlfors,
*On the fixed points of MĂ¶bius transformations in $\textbf {R}^n$*, Ann. Acad. Sci. Fenn. Ser. A I Math.**10**(1985), 15â27. MR**802464**, DOI 10.5186/aasfm.1985.1005 - Alan F. Beardon,
*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777**, DOI 10.1007/978-1-4612-1146-4 - S. S. Chen and L. Greenberg,
*Hyperbolic spaces*, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp.Â 49â87. MR**0377765** - C. Cao and P. L. Waterman,
*Conjugacy invariants of MĂ¶bius groups*, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995) Springer, New York, 1998, pp.Â 109â139. MR**1488448** - Leon Greenberg,
*Discrete subgroups of the Lorentz group*, Math. Scand.**10**(1962), 85â107. MR**141731**, DOI 10.7146/math.scand.a-10515 - James E. Humphreys,
*Linear algebraic groups*, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR**0396773**, DOI 10.1007/978-1-4684-9443-3 - Ravi S. Kulkarni,
*Dynamical types and conjugacy classes of centralizers in groups*, J. Ramanujan Math. Soc.**22**(2007), no.Â 1, 35â56. MR**2312547** - Ravi S. Kulkarni,
*Conjugacy classes in $M(n)$*, Conformal geometry (Bonn, 1985/1986) Aspects Math., E12, Friedr. Vieweg, Braunschweig, 1988, pp.Â 41â64. MR**979788** - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144** - John G. Ratcliffe,
*Foundations of hyperbolic manifolds*, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR**1299730**, DOI 10.1007/978-1-4757-4013-4 - Verma, Jugal K.,
*Invariants of finite groups*, preprint, http://www.math.iitb.ac.in/~jkv/, Pune (2000). - Masaaki Wada,
*Conjugacy invariants of MĂ¶bius transformations*, Complex Variables Theory Appl.**15**(1990), no.Â 2, 125â133. MR**1058518**, DOI 10.1080/17476939008814442 - P. L. Waterman,
*MĂ¶bius transformations in several dimensions*, Adv. Math.**101**(1993), no.Â 1, 87â113. MR**1239454**, DOI 10.1006/aima.1993.1043

## 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
- Received by editor(s): August 8, 2007
- Published electronically: March 26, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2491719