# DISCRETENESS BY USE OF A TEST MAP

• Li, Liulan (Department of Mathematics and Computational Science Hengyang Normal University) ;
• Fu, Xi (Department of Mathematics Hunan Normal University)
• Published : 2012.01.31

#### Abstract

It is well known that one could use a fixed loxodromic or parabolic element of a non-elementary group $G{\subset}M(\bar{\mathbb{R}}^n)$ as a test map to test the discreteness of G. In this paper, we show that a test map need not be in G. We also construct an example to show that the similar result using an elliptic element as a test map does not hold.

#### References

1. Y. Jiang, On the discreteness of Mobius groups in all dimensions, Math. Proc. Camb. Phi. Soc. 136 (2004), no. 3, 547-555. https://doi.org/10.1017/S0305004103007357
2. T. Jorgensen, On discrete groups of Mobius transformations, Amer. J. Math. 98 (1976), 739-749. https://doi.org/10.2307/2373814
3. T. Jorgensen, A note on subgroup of SL$(2,\mathbb{C})$, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 110, 209-211. https://doi.org/10.1093/qmath/28.2.209
4. L. Li and X. Wang, Discreteness criteria for Mobius groups acting on ${\bar{B}}^{n}$ II, Bull. Aust. Math. Soc. 80 (2009), no. 2, 275-290. https://doi.org/10.1017/S0004972709000264
5. P. Tukia, Differentiability and rigidity of Mobius groups, Invent. Math. 82 (1985), no.3, 557-578. https://doi.org/10.1007/BF01388870
6. P. Tukia and X. Wang, Discreteness of subgroups of SL$(2,\mathbb{C})$ containing elliptic elements, Math. Scand. 91 (2002), no. 2, 214-220. https://doi.org/10.7146/math.scand.a-14386
7. X. Wang, Dense subgroups of n-dimensional Mobius groups, Math. Z. 243 (2003), no. 4, 643-651. https://doi.org/10.1007/s00209-002-0437-3
8. X. Wang, L. Li, and W. Cao, Discreteness criteria for Mobius groups acting on ${\bar{B}}^{n}$, Israel J. Math. 150 (2005), 357-368. https://doi.org/10.1007/BF02762387
9. X. Wang and W. Yang, Discreteness criteria for subgroups in SL$(2,\mathbb{C})$, Math. Proc. Camb. Phi. Soc. 124 (1998), 51-55. https://doi.org/10.1017/S0305004197002387
10. X. Wang and W. Yang, Dense subgroups and discrete subgroups in SL$(2,\mathbb{C})$, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 517-521. https://doi.org/10.1093/qjmath/50.200.517
11. X. Wang and W. Yang, Generating systems of subgroups in PSL$(2,{\Gamma}_{n})$, Proc. Edinb. Math. Soc. (2) 45 (2002), no. 1, 49-58.
12. X. Wang and W. Yang, Discreteness criteria of Mobius groups of high dimensions and convergence theorems of Kleinian groups, Adv. Math. 159 (2001), no. 1, 68-82. https://doi.org/10.1006/aima.2000.1970
13. P. Waterman, Mobius transformations in several dimensions, Adv. Math. 101 (1993), no. 1, 87-113. https://doi.org/10.1006/aima.1993.1043
14. P. Waterman, Purely elliptic Mobius groups, Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986), 173-178, Math. Sci. Res. Inst. Publ., 11, Springer, New York, 1988.

#### Cited by

1. ON DISCRETENESS OF MÖBIUS GROUPS vol.50, pp.3, 2013, https://doi.org/10.4134/BKMS.2013.50.3.747