Bonfert-Taylor, Petra; Canary, Richard D.; Martin, Gaven; Taylor, Edward Quasiconformal homogeneity of hyperbolic manifolds. (English) Zbl 1063.30020 Math. Ann. 331, No. 2, 281-295 (2005). The first result proved here shows that there are severe restrictions on the geometry of uniformly quasiconformally homogeneous hyperbolic manifolds [cf. F. W. Gehring and B. P. Palka, J. Anal. Math. 30, 172–199 (1976; Zbl 0349.30019)]. In particular, a geometrically finite hyperbolic \(n\)-manifold is uniformly quasiconformally homogeneous if and only if it is closed. For a hyperbolic \(n\)-manifold with \(n > 2\), it is shown that (i) it is uniformly quasi-conformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold; (ii) there exists a uniform lower bound on the quasiconformal homogeneity constant. Deviations in the case of \(n=2\) are also highlighted. Reviewer: H. P. Dikshit (New Delhi) Cited in 3 ReviewsCited in 10 Documents MSC: 30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations Keywords:quasiconformal homogeneity; hyperbolic manifolds Citations:Zbl 0349.30019 PDFBibTeX XMLCite \textit{P. Bonfert-Taylor} et al., Math. Ann. 331, No. 2, 281--295 (2005; Zbl 1063.30020) Full Text: DOI References: [1] Apanasov, B.: Geometrically finite hyperbolic structures on manifolds. Ann. Global Anal. Geom. 1, 1-22 (1983) · Zbl 0531.57012 [2] Beardon, A., Maskit, B.: Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta Math. 132, 1-12 (1974) · Zbl 0277.30017 [3] Bonahon, F.: Bouts des variétiés hyperboliques de dimension 3. Ann. Math. 124, 71-158 (1986) · Zbl 0671.57008 [4] Bonfert-Taylor, P., Taylor, E.C.: Hausdorff dimension and limit sets of quasiconformal groups. Michigan Math. J. 49, 243-257 (2001) · Zbl 0999.30029 [5] Gehring, F.W., Martin, G.J.: Inequalities for Möbius transformations and discrete groups. J. Reine Angew. Math. 418, 31-76 (1991) · Zbl 0722.30034 [6] Gehring, F.W., Palka, B.: Quasiconformally homogeneous domains. J. Analyse Math. 30, 172-199 (1976) · Zbl 0349.30019 [7] Hempel, J.: 3-manifolds. Princeton University Press, 1976 · Zbl 0345.57001 [8] Hempel, J., Jaco, W.: Fundamental groups of 3-manifolds which are extensions. Ann. Math. 95, 86-98 (1972) · Zbl 0226.57003 [9] MacManus, P., Näkki, R., Palka, B.: Quasiconformally homogeneous compacta in the complex plane. Michigan Math. J. 45, 227-241 (1998) · Zbl 0960.30017 [10] MacManus, P., Näkki, R., Palka, B.: Quasiconformally bi-homogeneous compacta in the complex plane. Proc. London Math. Soc. 78, 215-240 (1999) · Zbl 0974.30012 [11] Malcev, A.: On faithful representations of infinite groups of matrices. Mat. Sb. 8(1940), 405-422, Am. Math. Soc. Translations 45, 1-18 (1965) · JFM 66.0088.03 [12] Martin, G.J.: On discrete Möbius groups in all dimensions: A generalization of Jørgensen?s inequality. Acta Math. 163, 253-289 (1989) · Zbl 0698.20037 [13] Maskit, B.: Kleinian Groups. Springer-Verlag, New York, 1987 · Zbl 0627.30039 [14] McMullen, C.T.: Renormalization and 3-Manifolds which Fiber over the Circle. Princeton University Press, 1996 · Zbl 0860.58002 [15] McMullen, C.T.: Complex earthquakes and Teichmüller theory. J. Am. Math. Soc. 11, 283-320 (1998) · Zbl 0890.30031 [16] Scott, P.: Compact submanifolds of 3-manifolds. J. London Math. Soc. 7, 246-250 (1974) · Zbl 0266.57001 [17] Selberg, A.: On discontinuous groups in higher dimensional symmetric spaces. In: Colloquium Function Theory, Tata Institute, 1960 · Zbl 0201.36603 [18] Thurston, W.P.: Three-dimensional Geometry and Topology. Princeton University Press, 1997 · Zbl 0873.57001 [19] Väisälä, J.: Lectures on n-dimensional Quasiconformal Mappings. Lecture Notes in Mathematics 229, Springer-Verlag, New York, 1971 · Zbl 0221.30031 [20] Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, 1319. Springer-Verlag, Berlin, 1988 · Zbl 0646.30025 [21] Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. 87, 56-88 (1968) · Zbl 0157.30603 [22] Wielenberg, N.: Discrete Moebius groups: fundamental polyhedra and convergence. Am. J. Math. 99, 861-877 (1977) · Zbl 0373.57024 [23] Yamada, A.: On Marden?s universal constant of Fuchsian groups. Kodai Math. J. 4, 266-277 (1981) · Zbl 0469.30038 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.