×

zbMATH — the first resource for mathematics

Test maps and discrete groups in SL\((2,C)\). (English) Zbl 1178.30024
Author’s abstract: We present several discreteness criterions for a non-elementary group \(G\) in SL\((2,C)\) by using a test map which need not be in \(G\).

MSC:
30C62 Quasiconformal mappings in the complex plane
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
PDF BibTeX XML Cite
Full Text: Euclid
References:
[1] A.F. Beardon: The Geometry of Discrete Groups, Graduate Texts in Mathematics 91 , Springer, New York, 1983. · Zbl 0528.30001
[2] M. Chen: Discreteness and convergence of Möbius groups , Geom. Dedicata 104 (2004), 61–69. · Zbl 1053.30038
[3] J. Gilman: Inequalities in discrete subgroups of \(\mathrm{PSL}(2,\mathbf{R})\) , Canad. J. Math. 40 (1988), 115–130. · Zbl 0629.20023
[4] N.A. Isokenko: Systems of generators of subgroups of \(\mathrm{PSL}(2,\mathbf{C})\) , Siberian Math. J. 31 (1990), 162–165.
[5] T. Jørgensen: On discrete groups of Möbius transformations , Amer. J. Math. 98 (1976), 739–749. JSTOR: · Zbl 0336.30007
[6] G. Rosenberger: Minimal generating systems of a subgroup of \(\mathrm{SL}(2,\mathbf{C})\) , Proc. Edinburgh Math. Soc. (2) 31 (1988), 261–265. · Zbl 0645.20030
[7] D. Sullivan: Quasiconformal homeomorphisms and dynamics: Structure stability implies hyperbolicity for Kleinian groups , Acta Math. 155 (1985), 243–260. · Zbl 0606.30044
[8] P. Tukia: Convergence groups and Gromov’s metric hyperbolic spaces , New Zealand J. Math. 23 (1994), 157–187. · Zbl 0855.30036
[9] P. Tukia and X. Wang: Discreteness of subgroups of \(\mathrm{SL}(2,\mathbf{C})\) containing elliptic elements , Math. Scand. 91 (2002), 214–220. · Zbl 1017.30053
[10] S. Yang: On the discreteness criterion in \(\mathrm{SL}(2,\mathbb{C})\) , Math. Z. 255 (2007), 227–230. · Zbl 1213.30047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.