The existence of quasimeromorphic mappings in dimension 3

Saucan, Emil

doi:10.1090/S1088-4173-06-00111-1

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-4173

The existence of quasimeromorphic mappings in dimension 3

Author: Emil Saucan
Journal: Conform. Geom. Dyn. 10 (2006), 21-40
MSC (2000): Primary 30C65, 57R05, 57M60
Posted: March 1, 2006
MathSciNet review: 2206314
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a Kleinian group acting on $\mathbb{H}^{3}$ admits a non-constant -automorphic function, even if it has torsion elements, provided that the orders of the elliptic elements are uniformly bounded. This is accomplished by developing a method for meshing distinct fat triangulations which is fatness preserving. We further show how to adapt the proof to higher dimensions.

[Ab] W. Abikoff, Kleinian Groups, lecture notes, The Technion--Israel Institute of Technology, Haifa, Israel, 1996-1997.
[Al] J.W. Alexander, Note on Riemmann spaces, Bull. Amer. Math. Soc. 26 (1920), 370-372.
[Ap] B. N. Apanasov, Kleinian groups in space, Sibirsk. Mat. Ž. 16 (1975), no. 5, 891–898, 1129 (Russian). MR 0404474 (53 #8276)
[Bea] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195 (97d:22011)
[Ber] Marcel Berger, Geometry. II, Universitext, Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy. MR 882916 (88a:51001b)
[BM] B. H. Bowditch and G. Mess, A 4-dimensional Kleinian group, Trans. Amer. Math. Soc. 344 (1994), no. 1, 391–405. MR 1240944 (95f:57057), http://dx.doi.org/10.1090/S0002-9947-1994-1240944-6
[BrM] Robert Brooks and J. Peter Matelski, Collars in Kleinian groups, Duke Math. J. 49 (1982), no. 1, 163–182. MR 650375 (83f:30039)
[Ca1] Stewart S. Cairns, On the triangulation of regular loci, Ann. of Math. (2) 35 (1934), no. 3, 579–587. MR 1503181, http://dx.doi.org/10.2307/1968752
[Ca2] Stewart S. Cairns, Polyhedral approximations to regular loci, Ann. of Math. (2) 37 (1936), no. 2, 409–415. MR 1503287, http://dx.doi.org/10.2307/1968452
[Ca3] Stewart S. Cairns, A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc. 67 (1961), 389–390. MR 0149491 (26 #6978), http://dx.doi.org/10.1090/S0002-9904-1961-10631-9
[CMS] Jeff Cheeger, Werner Müller, and Robert Schrader, On the curvature of piecewise flat spaces, Comm. Math. Phys. 92 (1984), no. 3, 405–454. MR 734226 (85m:53037)
[Cox] H. S. M. Coxeter, Regular polytopes, Second edition, The Macmillan Co., New York, 1963. MR 0151873 (27 #1856)
[DM] D. A. Derevnin and A. D. Mednykh, Geometric properties of discrete groups acting with fixed points in a Lobachevskiĭ space, Dokl. Akad. Nauk SSSR 300 (1988), no. 1, 27–30 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 3, 614–617. MR 948799 (90a:30131)
[FM] Mark Feighn and Geoffrey Mess, Conjugacy classes of finite subgroups of Kleinian groups, Amer. J. Math. 113 (1991), no. 1, 179–188. MR 1087807 (92a:57042), http://dx.doi.org/10.2307/2374827
[GM1] F. W. Gehring and G. J. Martin, Commutators, collars and the geometry of Möbius groups, J. Anal. Math. 63 (1994), 175–219. MR 1269219 (96c:30040), http://dx.doi.org/10.1007/BF03008423
[GM2] F. W. Gehring and G. J. Martin, On the Margulis constant for Kleinian groups. I, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 439–462. MR 1404096 (97f:30065)
[GMMR] F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3611–3643. MR 1433117 (98d:57022), http://dx.doi.org/10.1090/S0002-9947-97-01989-2
[H] Emily Hamilton, Geometrical finiteness for hyperbolic orbifolds, Topology 37 (1998), no. 3, 635–657. MR 1604903 (99h:57027), http://dx.doi.org/10.1016/S0040-9383(97)00043-8
[Hu] J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0248844 (40 #2094)
[J] Troels Jørgensen, On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), no. 3, 739–749. MR 0427627 (55 #658)
[KP] M. È. Kapovich and L. D. Potyagaĭlo, On the absence of finiteness theorems of Ahlfors and Sullivan for Kleinian groups in higher dimensions, Sibirsk. Mat. Zh. 32 (1991), no. 2, 61–73, 212 (Russian); English transl., Siberian Math. J. 32 (1991), no. 2, 227–237. MR 1138441 (93g:30064), http://dx.doi.org/10.1007/BF00972769
[KP1] Michael Kapovich and Leonid Potyagailo, On the absence of Ahlfors’ finiteness theorem for Kleinian groups in dimension three, Topology Appl. 40 (1991), no. 1, 83–91. MR 1114093 (92j:57023), http://dx.doi.org/10.1016/0166-8641(91)90060-Y
[Med] A.D. Mednikh, Automorphism groups of the three-dimensional hyperbolic manifolds, Soviet Math. Dokl. 32(3) (1985), 633-636.
[Mor] J.W. Morgan, On Thurston's Uniformization Theorem for Three-Dimensional Manifolds, in The Smith Conjecture (Morgan, J.W. and Bass, H. ed.), Academic Press, NY, 1984, 37-126.
[MS1] Olli Martio and Uri Srebro, Automorphic quasimeromorphic mappings in 𝑅ⁿ, Acta Math. 135 (1975), no. 3-4, 221–247. MR 0435388 (55 #8348)
[MS2] O. Martio and U. Srebro, On the existence of automorphic quasimeromorphic mappings in 𝑅ⁿ, Ann. Acad. Sci. Fenn. Ser. A I Math. 3 (1977), no. 1, 123–130. MR 0585312 (58 #28486)
[Ms] Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135 (90a:30132)
[My] Volker Mayer, Uniformly quasiregular mappings of Lattès type, Conform. Geom. Dyn. 1 (1997), 104–111 (electronic). MR 1482944 (98j:30017), http://dx.doi.org/10.1090/S1088-4173-97-00013-1
[Mun] James R. Munkres, Elementary differential topology, Lectures given at Massachusetts Institute of Technology, Fall, vol. 1961, Princeton University Press, Princeton, N.J., 1966. MR 0198479 (33 #6637)
[NW] Peter J. Nicholls and Peter L. Waterman, The boundary of convex fundamental domains of Fuchsian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 1, 11–25. MR 1050778 (91h:30066)
[Pe] Kirsi Peltonen, On the existence of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 85 (1992), 48. MR 1165363 (93h:30031)
[Rat] John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730 (95j:57011)
[S1] E. Saucan, The Existence of Quasimeromorphic Mappings, Ann. Acad. Sci. Fenn., Ser A I Math, 31, (2006), 131-142.
[S2] E. Saucan, Note on a theorem of Munkres, Mediterr. j. math 2(2) (2005), 215-229.
[S3] E. Saucan, in preparation.
[Som] D. M. Y. Sommerville, An introduction to the geometry of 𝑛 dimensions, Dover Publications Inc., New York, 1958. MR 0100239 (20 #6672)
[Spi V] Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, Publish or Perish Inc., Boston, Mass., 1975. MR 0394453 (52 #15254b)
[Sr] Uri Srebro, Non-existence of quasimeromorphic automorphic mappings, Analysis and topology, World Sci. Publ., River Edge, NJ, 1998, pp. 647–651. MR 1667838 (99j:30027)
[SA] E. Saucan and E. Apleboim, Quasiconformal Fold Elimination for Seaming and Tomography, in preparation.
[Th] William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975 (97m:57016)
[Tu] Pekka Tukia, Automorphic quasimeromorphic mappings for torsionless hyperbolic groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 545–560. MR 802519 (86k:30023)
[V] Jussi Väisälä, Lectures on 𝑛-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin, 1971. MR 0454009 (56 #12260)
[Wh] J. H. C. Whitehead, On 𝐶¹-complexes, Ann. of Math. (2) 41 (1940), 809–824. MR 0002545 (2,73d)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|