Notes on complex hyperbolic triangle groups

Kamiya, Shigeyasu; Parker, John; Thompson, James

doi:10.1090/S1088-4173-2010-00215-8

Notes on complex hyperbolic triangle groups
HTML articles powered by AMS MathViewer

by Shigeyasu Kamiya, John R. Parker and James M. Thompson

Conform. Geom. Dyn. 14 (2010), 202-218

DOI: https://doi.org/10.1090/S1088-4173-2010-00215-8

Published electronically: August 30, 2010

PDF | Request permission

Abstract:

We first demonstrate a family of isomorphisms between complex hyperbolic triangle groups and outline a systematic approach classifying the groups. Then we describe conditions that determine the discreteness of certain groups, in particular we prove a slightly weaker version of a conjecture given by Schwartz. Finally we collect together a list of known discrete triangle groups and propose some good candidates for discrete groups.

References

Martin Deraux, Deforming the $\Bbb R$-Fuchsian $(4,4,4)$-triangle group into a lattice, Topology 45 (2006), no. 6, 989–1020. MR 2263221, DOI 10.1016/j.top.2006.06.005
Martin Deraux, Elisha Falbel, and Julien Paupert, New constructions of fundamental polyhedra in complex hyperbolic space, Acta Math. 194 (2005), no. 2, 155–201. MR 2231340, DOI 10.1007/BF02393220
Elisha Falbel and John R. Parker, The geometry of the Eisenstein-Picard modular group, Duke Math. J. 131 (2006), no. 2, 249–289. MR 2219242, DOI 10.1215/S0012-7094-06-13123-X
E. Falbel, G. Francis, J.R. Parker, The geometry of the Gauss-Picard modular group. Preprint http://www.maths.dur.ac.uk/~dma0jrp/img/GaussPicard.pdf
William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR 1695450
S. Kamiya, J.R. Parker, J.M. Thompson, Non-Discrete Complex Hyperbolic Triangle Groups of Type $(n,n,\infty ;k)$. To appear in Canadian Mathematical Bulletin.
R.A. Livné, On certain covers of the universal elliptic curves. Ph.D. thesis, Havard University, 1981.
John R. Parker, Unfaithful complex hyperbolic triangle groups. I. Involutions, Pacific J. Math. 238 (2008), no. 1, 145–169. MR 2443511, DOI 10.2140/pjm.2008.238.145
John R. Parker, Cone metrics on the sphere and Livné’s lattices, Acta Math. 196 (2006), no. 1, 1–64. MR 2237205, DOI 10.1007/s11511-006-0001-9
Anna Pratoussevitch, Traces in complex hyperbolic triangle groups, Geom. Dedicata 111 (2005), 159–185. MR 2155180, DOI 10.1007/s10711-004-1493-0
Richard Evan Schwartz, Complex hyperbolic triangle groups, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 339–349. MR 1957045
Richard Evan Schwartz, Real hyperbolic on the outside, complex hyperbolic on the inside, Invent. Math. 151 (2003), no. 2, 221–295. MR 1953259, DOI 10.1007/s00222-002-0245-8
J.M. Thompson, Two new triangle group lattices in the complex hyperbolic plane. Preprint http://maths.dur.ac.uk/~dma2jmt/TwoNewLattices.pdf
J. Wyss-Gallifent, Complex Hyperbolic Triangle Groups. Ph.D. thesis, University of Maryland, 2000.