Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T16:16:18.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Global Geometrical Coordinates on Falbel's Cross-Ratio Variety

Published online by Cambridge University Press:  20 November 2018

John R. Parker  and
Ioannis D. Platis
John R. Parker
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom e-mail: [email protected]
Ioannis D. Platis
Affiliation:
Department of Mathematics, Aristotle University of Salonica, Salonica, Greece e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Falbel has shown that four pairwise distinct points on the boundary of a complex hyperbolic 2-space are completely determined, up to conjugation in $\text{PU}\left( 2,\,1 \right)$, by three complex cross-ratios satisfying two real equations. We give global geometrical coordinates on the resulting variety.

Keywords

32G0532M05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 2 , 01 June 2009 , pp. 285 - 294
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Beardon, A. F., The geometry of discrete groups. Graduate Texts in Mathematics 91, Springer-Verlag, New York, 1983.Google Scholar
[2] Cunha, H. and Gusevskii, N., On the moduli space of quadruples of points in the boundary of complex hyperbolic space. arXiv:0812.2159v1[math.GT] 11 Dec. 2008.Google Scholar
[3] Falbel, E., Geometric structures associated to triangulations as fixed point sets of involutions. Topology Appl. 154(2007), no. 6, 1041–1052.Google Scholar
[4] Falbel, E. and Platis, I. D., The PU(2, 1) configuration space of four points in S 3 and the cross-ratio variety. Math. Ann. 340(2008), 935–962.Google Scholar
[5] Fenchel, W., Elementary geometry in hyperbolic space. de Gruyter Studies in Mathematics 11, Walter de Gruyter and Co., Berlin, 1989.Google Scholar
[6] Goldman, W. M., Complex hyperbolic geometry. Oxford Mathematical Monographs, Oxford University Press, New York, 1999.Google Scholar
[7] Jiang, Y., Kamiya, S., and Parker, J. R., Jørgensen's inequality for complex hyperbolic space. Geom. Dedicata 97(2003), 55–80.Google Scholar
[8] Korányi, A. and Reimann, H. M., The complex cross ratio on the Heisenberg group. Enseign. Math. 33(1987), no. 3–4, 291–300.Google Scholar
[9] Kourouniotis, C., Complex length coordinates for quasi-Fuchsian groups. Mathematika 41(1994), no. 1, 173–188.Google Scholar
[10] Parker, J. R. and Platis, I. D.; Complex hyperbolic Fenchel–Nielsen coordinates. Topology 47(2008), no. 2, 101–135.Google Scholar
[11] Parker, J. R. and Series, C., Bending formulae for convex hull boundaries. J. Anal. Math. 67(1995), 165–198.Google Scholar
[12] Will, P., Groupes libres, groupes triangulaires et tore épointé dans PU(2, 1). Ph.D. thesis, University of Paris VI, 2006.Google Scholar