Proceedings of the Japan Academy, Series A, Mathematical Sciences

Note on non-discrete complex hyperbolic triangle groups of type $(n,n,\infty;k)$

Shigeyasu Kamiya

Abstract

A complex hyperbolic triangle group is a group generated by three complex involutions fixing complex lines in complex hyperbolic space. In a previous paper~[3] we discussed complex hyperbolic triangle groups of type $(n,n,\infty;k)$ and proved that for $n \geq 29$ these groups are not discrete. In this paper we show that if $n \geq 22$, then complex hyperbolic triangle groups of type $(n,n,\infty;k)$ are not discrete and give a new list of non-discrete groups of type $(n,n,\infty;k)$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 89, Number 8 (2013), 100-102.

Dates
First available in Project Euclid: 17 October 2013

https://projecteuclid.org/euclid.pja/1382016181

Digital Object Identifier
doi:10.3792/pjaa.89.100

Mathematical Reviews number (MathSciNet)
MR3127925

Zentralblatt MATH identifier
1294.22007

Citation

Kamiya, Shigeyasu. Note on non-discrete complex hyperbolic triangle groups of type $(n,n,\infty;k)$. Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 8, 100--102. doi:10.3792/pjaa.89.100. https://projecteuclid.org/euclid.pja/1382016181

References

• J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arith. 30 (1976), no. 3, 229–240.
• W. M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, Oxford Univ. Press, New York, 1999.
• S. Kamiya, J. R. Parker and J. M. Thompson, Non-discrete complex hyperbolic triangle groups of type $(n,n,\infty;k)$, Canad. Math. Bull. 55 (2012), no. 2, 329–338.
• S. Kamiya, J. R. Parker and J. M. Thompson, Notes on complex hyperbolic triangle groups, Conform. Geom. Dyn. 14 (2010), 202–218.
• R. E. Schwartz, Complex hyperbolic triangle groups, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 339–349, Higher Ed. Press, Beijing, 2002.