Osaka Journal of Mathematics

Discreteness criterion in $\mathrm{SL}(2,\mathbb{C})$ by a test map

Wensheng Cao

Abstract

In the paper [12], Yang conjectured that a nonelementary subgroup $G$ of $\mathrm{SL}(2, \mathbb{C})$ containing elliptic elements is discrete if for each elliptic element $g \in G$ the group $\langle f, g \rangle$ is discrete, where $f \in \mathrm{SL}(2,\mathbb{C})$ is a test map being loxodromic or elliptic. By embedding $\mathrm{SL}(2,\mathbb{C})$ into $\mathrm{U}(1,1; \mathbb{H})$, we give an affirmative answer to this question. As an application, we show that a nonelementary and nondiscrete subgroup of $\mathrm{Isom}(H^{3})$ must contain an elliptic element of order at least 3.

Article information

Source
Osaka J. Math., Volume 49, Number 4 (2012), 901-907.

Dates
First available in Project Euclid: 19 December 2012

https://projecteuclid.org/euclid.ojm/1355926881

Mathematical Reviews number (MathSciNet)
MR3007948

Zentralblatt MATH identifier
1294.30035

Citation

Cao, Wensheng. Discreteness criterion in $\mathrm{SL}(2,\mathbb{C})$ by a test map. Osaka J. Math. 49 (2012), no. 4, 901--907. https://projecteuclid.org/euclid.ojm/1355926881

References

• W. Cao, J.R. Parker and X. Wang: On the classification of quaternionic Möbius transformations, Math. Proc. Cambridge Philos. Soc. 137 (2004), 349–361.
• W. Cao and H. Tan: Jørgensen's inequality for quaternionic hyperbolic space with elliptic elements, Bull. Aust. Math. Soc. 81 (2010), 121–131.
• M. Chen: Discreteness and convergence of Möbius groups, Geom. Dedicata 104 (2004), 61–69.
• S.S. Chen and L. Greenberg: Hyperbolic spaces; in Contributions to Analysis, Academic Press, New York, 1974, 49–87.
• J. Gilman: Inequalities in discrete subgroups of $\PSL(2,\mathbf{R})$, Canad. J. Math. 40 (1988), 115–130.
• L. Greenberg: Discrete subgroups of the Lorentz group, Math. Scand. 10 (1962), 85–107.
• N.A. Isachenko: Systems of generators of subgroups of $\PSL(2,\mathbf{C})$, Siberian Math. J. 31 (1990), 162–165.
• \begingroup T. Jørgensen: On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), 739–749. \endgroup
• D. Sullivan: Quasiconformal homeomorphisms and dynamics, II, Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), 243–260.
• P. Tukia and X. Wang: Discreteness of subgroups of $\SL(2,\mathbf{C})$ containing elliptic elements, Math. Scand. 91 (2002), 214–220.
• S. Yang: On the discreteness criterion in $\SL(2,\mathbb{C})$, Math. Z. 255 (2007), 227–230.
• S. Yang: Test maps and discrete groups in $\mathit{SL}(2,C)$, Osaka J. Math. 46 (2009), 403–409.
• S. Yang: Elliptic elements in Möbius groups, Israel J. Math. 172 (2009), 309–315. \endthebibliography*