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discrete group; geometric convergence; uniformly bounded torsion
One of the basic questions in the Kleinian group theory is to understand both algebraic and geometric limiting behavior of sequences of discrete subgroups. In this paper we consider the geometric convergence in the setting of the isometric group of the real or complex hyperbolic space. It is known that if $\Gamma $ is a non-elementary finitely generated group and $\rho _{i}\colon \Gamma \rightarrow {\rm SO}(n,1)$ a sequence of discrete and faithful representations, then the geometric limit of $\rho _{i}(\Gamma )$ is a discrete subgroup of ${\rm SO}(n,1)$. We generalize this result by showing that for a sequence of discrete and non-elementary subgroups $\{G_{j}\}$ of ${\rm SO}(n,1)$ or ${\rm PU}(n,1)$, if $\{G_{j}\}$ has uniformly bounded torsion, then its geometric limit is either elementary, or discrete and non-elementary.
[1] Abikoff, W., Haas, A.: Nondiscrete groups of hyperbolic motions. Bull. Lond. Math. Soc. 22 (1990), 233-238. DOI 10.1112/blms/22.3.233 | MR 1041136 | Zbl 0709.20026
[2] Belegradek, I.: Intersections in hyperbolic manifolds. Geom. Topol. 2 (1998), 117-144. DOI 10.2140/gt.1998.2.117 | MR 1633286 | Zbl 0931.57009
[3] Chen, S. S., Greenberg, L.: Hyperbolic spaces. Contributions to Analysis, Collection of Papers Dedicated to Lipman Bers L. V. Ahlfors et al. Academic Press, New York (1974), 49-87. MR 0377765 | Zbl 0295.53023
[4] Jørgensen, T., Marden, A.: Algebraic and geometric convergence of Kleinian groups. Math. Scand. 66 (1990), 47-72. DOI 10.7146/math.scand.a-12292 | MR 1060898 | Zbl 0738.30032
[5] Kapovich, M.: Hyperbolic Manifolds and Discrete Groups. Reprint of the 2001 edition. Modern Birkhäuser Classics Birkhäuser, Boston (2009). MR 2553578
[6] Martin, G. J.: On discrete isometry groups of negative curvature. Pac. J. Math. 160 (1993), 109-127. DOI 10.2140/pjm.1993.160.109 | MR 1227506 | Zbl 0822.57026
[7] Martin, G. J.: On discrete Möbius groups in all dimensions: A generalization of Jørgensen's inequality. Acta Math. 163 (1989), 253-289. DOI 10.1007/BF02392737 | MR 1032075 | Zbl 0698.20037
[8] Tukia, P.: Convergence groups and Gromov's metric hyperbolic spaces. N. Z. J. Math. (electronic only) 23 (1994), 157-187. MR 1313451 | Zbl 0855.30036
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