Subgroups of 𝑃𝑆𝐿(3,ℂ) with four lines in general position in its limit set

Barrera, W.; Cano, A.; Navarrete, J.

doi:10.1090/S1088-4173-2011-00231-1

Subgroups of $\text \textit {PSL}(3,\mathbb {C})$ with four lines in general position in its limit set
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by W. Barrera, A. Cano and J. P. Navarrete

Conform. Geom. Dyn. 15 (2011), 160-176

DOI: https://doi.org/10.1090/S1088-4173-2011-00231-1

Published electronically: October 11, 2011

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Abstract:

In this article we provide an algebraic characterization of the subgroups of $PSL(3,\Bbb {C})$ for which the maximum number of complex lines in general position contained in its limit set, according to Kulkarni, is equal to four. Also, we give an explicit description of the discontinuity region, according to Kulkarni, of such groups.

References

Waldemar del Jesús Barrera Vargas, Angel Cano Cordero, and Juan Pablo Navarrete Carrillo, The limit set of discrete subgroups of $\textrm {PSL}(3,\Bbb C)$, Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 1, 129–146. MR 2739076, DOI 10.1017/S0305004110000423
A. Cano, On Discrete Subgroups of automorphism of $\mathbb {P}^2_{\mathbb {C}}$, preprint, 2009, http://arxiv.org/ abs/0806.1336.
A. Cano, J. Seade, J. P. Navarrete, Complex Kleinian Groups, preprint, 2009, http://www. matcuer.unam.mx/\symbol{126}jseade/Libro-Klein.pdf.
Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
É. Ghys, A. Haefliger, and A. Verjovsky (eds.), Group theory from a geometrical viewpoint, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1170362, DOI 10.1142/1235
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
R. S. Kulkarni, Groups with domains of discontinuity, Math. Ann. 237 (1978), no. 3, 253–272. MR 508756, DOI 10.1007/BF01420180
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
J.-P. Navarrete, On the limit set of discrete subgroups of $\textrm {PU}(2,1)$, Geom. Dedicata 122 (2006), 1–13. MR 2295538, DOI 10.1007/s10711-006-9051-6
John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730, DOI 10.1007/978-1-4757-4013-4