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Non-Discrete Complex Hyperbolic Triangle Groups of Type (n, n, ∞; k)

Published online by Cambridge University Press:  20 November 2018

Shigeyasu Kamiya ,
John R. Parker  and
James M. Thompson
Shigeyasu Kamiya
Affiliation:
Okayama University of Science, 1-1 Ridai-cho, Okayama 700-0005, Japane-mail: s.kamiya@are.ous.ac.jp
John R. Parker
Affiliation:
University of Durham, South Road, Durham DH1 3LE UKe-mail: j.r.parker@dur.ac.uk; j.m.thompson@dur.ac.uk
James M. Thompson
Affiliation:
University of Durham, South Road, Durham DH1 3LE UKe-mail: j.r.parker@dur.ac.uk; j.m.thompson@dur.ac.uk
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Abstract

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A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,\,n,\,\infty ;\,k)$.

Keywords

51M1032M1553C5553C35complex hyperbolic triangle group

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 329 - 338
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Conway, J. H. and Jones, A. J., Trigonometric Diophantine equations. (On vanishing sums of roots of unity). Acta Arith. 30(1976), no. 3, 229–240.Google Scholar
[2] Goldman, W. M., Complex Hyperbolic Geometry. Oxford University Press, New York, 1999.Google Scholar
[3] Goldman, W. M. and Parker, J. R., Complex hyperbolic ideal triangle groups. J. Reine Angew. Math. 425(1992), 71–86.Google Scholar
[4] Kamiya, S., On discrete subgroups of PU(1, 2; C) with Heisenberg translations. J. London Math. Soc. 62(2000), no. 3, 827–842. http://dx.doi.org/10.1112/S0024610700001435 Google Scholar
[5] Kamiya, S., Remarks on complex hyperbolic triangle groups. In: Complex Analysis and Its Applictions. OCAMI Stud. 2. Osaka Munic. Univ. Press, Osaka, 2007, pp. 219–223.Google Scholar
[6] Kamiya, S. and Parker, J. R., Discrete subgroups of PU(2, 1) with screw parabolic elements. Math. Proc. Cambridge Philos. Soc. 144(2008), no. 2, 443–455.Google Scholar
[7] Parker, J. R., Uniform discreteness and Heisenberg translation. Math. Z. 225(1997), no. 3, 485–505. http://dx.doi.org/10.1007/PL00004315 Google Scholar
[8] Parker, J. R., Unfaithful complex hyperbolic triangle groups. I: Involutions. Pacific J. Math. 238(2008), no. 1, 145–169. http://dx.doi.org/10.2140/pjm.2008.238.145 Google Scholar
[9] Pratoussevitch, A., Traces in complex hyperbolic triangle groups. Geom. Dedicata 111(2005), 159–185. http://dx.doi.org/10.1007/s10711-004-1493-0 Google Scholar
[10] Pratoussevitch, A., Non-discrete complex hyperbolic triangle groups of type (m, m, ∞) . Bull. London Math. Soc. 43(2011), no. 2, 359–363. http://dx.doi.org/10.1112/blms/bdq107 Google Scholar
[11] Schwartz, R. E., Ideal triangle groups, dented tori, and numerical analysis. Ann. of Math. 153(2001), no. 3, 533–598. http://dx.doi.org/10.2307/2661362 Google Scholar
[12] Schwartz, R. E., Degenerating the complex hyperbolic ideal triangle groups. Acta Math. 186(2001), no. 1, 105–154. http://dx.doi.org/10.1007/BF02392717 Google Scholar
[13] Schwartz, R. E., Complex hyperbolic triangle groups. Proceedings of International Congress of Mathematicians, Vol. II. Higher Ed. Press, Beijing, 2002, pp. 339–349.Google Scholar
[14] Wyss-Gallifent, J., Complex Hyperbolic Triangle Groups. Ph.D. thesis, University of Maryland, 2000.Google Scholar
[15] Xie, B.-H. and Jiang, Y.-P., Discreteness of subgroups of PU(2, 1) with regular elliptic elements. Linear Algebra Appl. textbf428(2008), no. 4, 732–737. http://dx.doi.org/10.1016/j.laa.2007.07.021 Google Scholar