Abstract
Poincaré’s polyhedron theorem gives geometrical conditions on a domain constructed with spherical sides so that the group generated by some elements which permute those sides is discrete. The polyhedron we construct in complex hyperbolic space is bounded by bisectors. We will see a particular form originally proposed by Mostow, and will prove it in the same fashion with real hyperbolic case. Then we will apply it to investigation of the discreteness of complex hyperbolic ultra-ideal triangle groups.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Beardon, A.F.: The geometry of discrete groups, Graduate Texts in Mathematics (1983)
Cao, W., Parker, J.R.: Jørgensen’s inequality and collars in \(n\)-dimensional quaternionic hyperbolic space. Q. J. Math. 62, 523–543 (2011)
Deraux, M., Parker, J.R., Paupert, J.: New non-arithmetic complex hyperbolic lattices. Invent. Math. 203, 681–771 (2015)
Epstein, D.B.A., Petronio, C.: An exposition of Poincaré’s polyhedron theorem, Enseign. Math. 40(2), 113–170 (1994)
Falbel, E., Zocca, V.: A Poincaré’s polyhedron theorem for complex hyperbolic geometry. J. Reine Angew. Math. 516, 133–158 (1999)
Goldman, W.M.: Complex Hyperbolic Geometry, Oxford University Press (1999)
Gusevskii, N., Parker, J.R.: Representations of free Fuchsian groups in complex hyperbolic space. Topology 39, 33–60 (2000)
Jiang, Y., Kamiya, S., Parker, J.R.: Jørgensen’s inequality for complex hyperbolic space. Geom. Dedicata, 97 (2003), pp. 55–80. Special volume dedicated to the memory of Hanna Miriam Sandler (1960–1999)
Kim, I., Parker, J.R.: Geometry of quaternionic hyperbolic manifolds. Math. Proc. Cambridge Philos. Soc. 135, 291–320 (2003)
Maskit, B.: Kleinian groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287. Springer-Verlag, Berlin (1988)
Parker, J.R.: Complex hyperbolic keinian groups. Preprint
Parker, J.R.: Shimizu’s lemma for complex hyperbolic space. Internat. J. Math. 3, 291–308 (1992)
Parker, J.R., Wang, J., Xie, B.: Complex hyperbolic \((3,3, n)\) triangle groups. Pacific J. Math. 280, 433–453 (2016)
Schwartz, R.E.: Degenerating the complex hyperbolic ideal triangle groups. Acta Math. 186, 105–154 (2001)
Acknowledgements
This article was a part of the author’s PhD thesis and she would like to thank Professor Toshiyuki Sugawa for his patient guidance and valuable suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Sun, LJ. (2018). A Note on Poincaré’s Polyhedron Theorem in Complex Hyperbolic Space. In: Byun, J., Cho, H., Kim, S., Lee, KH., Park, JD. (eds) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol 246. Springer, Singapore. https://doi.org/10.1007/978-981-13-1672-2_25
Download citation
DOI: https://doi.org/10.1007/978-981-13-1672-2_25
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-1671-5
Online ISBN: 978-981-13-1672-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)