Algebraic characterization of isometries of the complex and the quaternionic hyperbolic 3-spaces

Gongopadhyay, Krishnendu

doi:10.1090/S0002-9939-2012-11422-4

Algebraic characterization of isometries of the complex and the quaternionic hyperbolic $3$-spaces
HTML articles powered by AMS MathViewer

by Krishnendu Gongopadhyay PDF

Proc. Amer. Math. Soc. 141 (2013), 1017-1027 Request permission

Abstract:

Let $\mathbf {H}^3_{\mathbb {F}}$ denote the three dimensional hyperbolic space over $\mathbb {F}$, where $\mathbb {F}$ denotes either the complex numbers $\mathbb {C}$ or the quaternions $\mathbb {H}$. We offer an algebraic characterization of isometries of $\mathbf {H}^3_{\mathbb {F}}$.

References

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
W. S. Burnside and A. W. Patron, The theory of equations with an introduction to the theory of binary algebraic forms, Dublin University Press Series, 1881.
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
L. E. Dickson, Elementary Theory of Equations. John Wiley & Sons, New York, 1914.
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
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
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
Juan-Pablo Navarrete, The trace function and complex Kleinian groups in $\Bbb P^2_{\Bbb C}$, Internat. J. Math. 19 (2008), no. 7, 865–890. MR 2437075, DOI 10.1142/S0129167X08004868
E. L. Rees, Graphical Discussion of the Roots of a Quartic Equation, Amer. Math. Monthly 29 (1922), no. 2, 51–55. MR 1519936, DOI 10.2307/2972804
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
José Seade and Alberto Verjovsky, Actions of discrete groups on complex projective spaces, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) Contemp. Math., vol. 269, Amer. Math. Soc., Providence, RI, 2001, pp. 155–178. MR 1810539, DOI 10.1090/conm/269/04332
Fuzhen Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997), 21–57. MR 1421264, DOI 10.1016/0024-3795(95)00543-9