Classification of quaternionic hyperbolic isometries

Gongopadhyay, Krishnendu; Parsad, Shiv

doi:10.1090/S1088-4173-2013-00256-7

Classification of quaternionic hyperbolic isometries
HTML articles powered by AMS MathViewer

by Krishnendu Gongopadhyay and Shiv Parsad

Conform. Geom. Dyn. 17 (2013), 68-76

DOI: https://doi.org/10.1090/S1088-4173-2013-00256-7

Published electronically: May 6, 2013

PDF | Request permission

Abstract:

Let $\mathbb {F}$ denote either the complex numbers $\mathbb {C}$ or the quaternions $\mathbb {H}$. Let $\mathbf {H}_{\mathbb {F}}^n$ denote the $n$-dimensional hyperbolic space over $\mathbb {F}$. We obtain algebraic criteria to classify the isometries of $\mathbf {H}_{\mathbb {F}}^n$. This generalizes the work in Geom. Dedicata 157 (2012), 23–39 and Proc. Amer. Math. Soc. 141 (2013), 1017–1027, to isometries of arbitrary dimensional quaternionic hyperbolic space. As a corollary, a characterization of isometries of $\mathbf {H}_{\mathbb {C}}^n$ is also obtained.

References

Lars V. Ahlfors, On the fixed points of Möbius transformations in $\textbf {R}^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 15–27. MR 802464, DOI 10.5186/aasfm.1985.1005
Daniel Allcock, New complex- and quaternion-hyperbolic reflection groups, Duke Math. J. 103 (2000), no. 2, 303–333. MR 1760630, DOI 10.1215/S0012-7094-00-10326-2
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
Wensheng Cao and Krishnendu Gongopadhyay, Commuting isometries of the complex hyperbolic space, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3317–3326. MR 2811286, DOI 10.1090/S0002-9939-2011-10796-2
Wensheng Cao and Krishnendu Gongopadhyay, Algebraic characterization of isometries of the complex and the quaternionic hyperbolic planes, Geom. Dedicata 157 (2012), 23–39. MR 2893478, DOI 10.1007/s10711-011-9599-7
Wensheng Cao and John R. Parker, Jørgensen’s inequality and collars in $n$-dimensional quaternionic hyperbolic space, Q. J. Math. 62 (2011), no. 3, 523–543. MR 2825470, DOI 10.1093/qmath/haq003
Wensheng Cao, John R. Parker, and Xiantao Wang, On the classification of quaternionic Möbius transformations, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2, 349–361. MR 2092064, DOI 10.1017/S0305004104007868
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. 49–87. MR 0377765
R. Dye and R. W. D. Nickalls, The geometry of the discriminant of a polynomial. The Mathematical Gazette, vol. 80, no. 488 (1996), 279-285.
Georges Giraud, Sur certaines fonctions automorphes de deux variables, Ann. Sci. École Norm. Sup. (3) 38 (1921), 43–164 (French). MR 1509233, DOI 10.24033/asens.731
William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR 1695450
Krishnendu Gongopadhyay, Algebraic characterization of the isometries of the hyperbolic 5-space, Geom. Dedicata 144 (2010), 157–170. MR 2580424, DOI 10.1007/s10711-009-9394-x
Krishnendu Gongopadhyay, Algebraic characterization of isometries of the complex and the quaternionic hyperbolic $3$-spaces, Proc. Amer. Math. Soc. 141 (2013), no. 3, 1017–1027. MR 3003693, DOI 10.1090/S0002-9939-2012-11422-4
K. Gongopadhyay, The z-classes of quaternionic hyperbolic isometries, J. Group Theory, DOI 10.1515/jgt-2013-0013.
K. Gongopadhyay, J. R. Parker and S. Parsad, On the classification of unitary matrices, Preprint.
Lu Yang, Xiaorong Hou, and Zhenbing Zeng, A complete discrimination system for polynomials, Sci. China Ser. E 39 (1996), no. 6, 628–646. MR 1438754
Inkang Kim and John R. Parker, Geometry of quaternionic hyperbolic manifolds, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 2, 291–320. MR 2006066, DOI 10.1017/S030500410300687X
H. C. Lee, Eigenvalues and canonical forms of matrices with quaternion coefficients, Proc. Roy. Irish Acad. Sect. A 52 (1949), 253–260. MR 0036738
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
G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
S. Wolfram, “Mathematica: A System for Doing Mathematics by Computer,” Wolfram Press, 2000.
Masaaki Wada, Conjugacy invariants of Möbius transformations, Complex Variables Theory Appl. 15 (1990), no. 2, 125–133. MR 1058518, DOI 10.1080/17476939008814442
P. L. Waterman, Möbius transformations in several dimensions, Adv. Math. 101 (1993), no. 1, 87–113. MR 1239454, DOI 10.1006/aima.1993.1043
Wolfram Demonstration Project, The structure of real roots of a quintic polynomial, http://demonstrations.wolfram.com/TheStructureOfTheRealRootsOsAQuinticPolynomial/ Contributed by: Andrej Kozlowski.
Fuzhen Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997), 21–57. MR 1421264, DOI 10.1016/0024-3795(95)00543-9