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Geometriae Dedicata

, 129:119 | Cite as

Aspherical manifolds with relatively hyperbolic fundamental groups

  • Igor Belegradek
Original Paper

Abstract

We show that the aspherical manifolds produced via the relative strict hyperboli- zation of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, and acylindricity. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental groups of boundary components. We also show that no manifold obtained via the relative strict hyperbolization can be embedded into a compact Kähler manifold of the same dimension, except when the dimension is two.

Keywords

Relatively hyperbolic Hyperbolization of polyhedra Aspherical manifold Splitting Simplicial volume Co-Hopfian Kähler 

Mathematics Subject Classification (2000)

20F65 

References

  1. Alexander S.B. and Bishop R.L. (2005). A cone splitting theorem for Alexandrov spaces. Pacific J. Math. 218(1): 1–15 MATHMathSciNetCrossRefGoogle Scholar
  2. Amorós, J., Burger M., Corlette K., Kotschick D., Toledo D.: Fundamental groups of compact Kähler manifolds. In: Mathematical Surveys and Monographs, vol. 44. American Mathematical Society, Providence, RI (1996)Google Scholar
  3. Behrstock J., Drutu, C., Mosher L.: Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. arXiv:math.GT/0512592Google Scholar
  4. Belegradek I. (2002). On Mostow rigidity for variable negative curvature. Topology 41(2): 341–361 MATHCrossRefMathSciNetGoogle Scholar
  5. Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry. Universitext, Springer-Verlag (1992)Google Scholar
  6. Bestvina M. and Feighn M. (1995). Stable actions of groups on real trees. Invent. Math. 121(2): 287–321 MATHCrossRefMathSciNetGoogle Scholar
  7. Bowditch, B.H.: Relatively Hyperbolic Groups. Southampton preprint (1999). http://www.maths.soton.ac.uk/staff/Bowditch/preprints.html. Accessed 3 Jan 2007
  8. Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag (1999)Google Scholar
  9. Burger M. and Monod N. (1999). Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. (JEMS) 1(2): 199–235 MATHCrossRefMathSciNetGoogle Scholar
  10. Burger M. and Monod N. (1999). Erratum: Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. (JEMS) 1(3): 338 CrossRefMathSciNetGoogle Scholar
  11. Canary R.D. and McCullough D. (2004). Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups. Mem. Amer. Math. Soc. 172(812): xii+218 MathSciNetGoogle Scholar
  12. Canary, R.D., Epstein, D.B.A., Green, P.: Notes on notes of Thurston. In: Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 111, pp. 3–92. Cambridge Univ. Press, Cambridge (1987)Google Scholar
  13. Charney R.M. and Davis M.W. (1995). Strict hyperbolization. Topology 34(2): 329–350 MATHCrossRefMathSciNetGoogle Scholar
  14. Davis, M.W.: Exotic aspherical manifolds. In: Lueck, W., Farrell, F.T. (eds.) Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), ICTP Lect. Notes, vol. 9, pp. 371–404. Abdus Salam Int. Cent. Theoret. Phys., Trieste (2002)Google Scholar
  15. Davis M.W. and Januszkiewicz T. (1991). Hyperbolization of polyhedra. J. Differential Geom. 34(2): 347–388 MATHMathSciNetGoogle Scholar
  16. Davis, M.W., Januszkiewicz, T., Weinberger, S.: Relative hyperbolization and aspherical bordisms: an addendum to “Hyperbolization of polyhedra” [J. Differential Geom. 34(1991), (2), 347–388; MR1131435 (92h:57036)] by Davis and Januszkiewicz, J. Differential Geom. 58(3), 535–541 (2001)Google Scholar
  17. de la Harpe, P., Valette, A.: La propriété (T) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger). Astérisque 175, (1989), With an appendix by M. BurgerGoogle Scholar
  18. Delzant, T., Gromov M.: Cuts in Kähler groups. In: Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., vol. 248, pp. 31–55. Birkhäuser, Basel (2005)Google Scholar
  19. Deraux M. (2005). A negatively curved Kähler threefold not covered by the ball. Invent. Math. 160(3): 501–525 MATHCrossRefMathSciNetGoogle Scholar
  20. Druţu, C., Sapir, M.: Groups acting on tree-graded spaces and splittings of relatively hyperbolic group. arXiv:math.GR/0601305Google Scholar
  21. Druţu, C., Sapir M.: Tree-graded spaces and asymptotic cones of groups. Topology 44(5), 959–1058 (2005), With an appendix by D. Osin and SapirGoogle Scholar
  22. Farb B. (1998). Relatively hyperbolic groups. Geom. Funct. Anal. 8(5): 810–840 MATHCrossRefMathSciNetGoogle Scholar
  23. Farrell F.T. and Jones L.E. (1989). Negatively curved manifolds with exotic smooth structures. J. Amer. Math. Soc. 2(4): 899–908 MATHCrossRefMathSciNetGoogle Scholar
  24. Farrell F.T., Jones L.E. and Ontaneda P. (1998). Hyperbolic manifolds with negatively curved exotic triangulations in dimensions greater than five. J. Differential Geom. 48(2): 319–322 MATHMathSciNetGoogle Scholar
  25. Francaviglia, S.: Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds. Int. Math. Res. Not. (9), 425–459 (2004)Google Scholar
  26. Fujiwara K. (1998). The second bounded cohomology of a group acting on a Gromov-hyperbolic space. Proc. London Math. Soc. (3) 76(1): 70–94 CrossRefMathSciNetGoogle Scholar
  27. Goldfarb B. (1999). Novikov conjectures and relative hyperbolicity. Math. Scand. 85(2): 169–183 MATHMathSciNetGoogle Scholar
  28. Gromov, M., Piatetski-Shapiro, I.: Nonarithmetic groups in Lobachevsky spaces. Inst. Hautes Études Sci. Publ. Math. (66), 93–103 (1988)Google Scholar
  29. Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. (1982) (56), 5–99 (1983).Google Scholar
  30. Gromov, M.: Hyperbolic groups, Essays in group theory. Math. Sci. Res. Inst. Publ., vol. 8, pp. 75–263. Springer (1987)Google Scholar
  31. Gromov, M., Schoen, R.: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. (76), 165–246 (1992)Google Scholar
  32. Gromov M. and Thurston W. (1987). Pinching constants for hyperbolic manifolds. Invent. Math. 89(1): 1–12 MATHCrossRefMathSciNetGoogle Scholar
  33. Hamrick G.C. and Royster D.C. (1982). Flat Riemannian manifolds are boundaries. Invent. Math. 66(3): 405–413 MATHCrossRefMathSciNetGoogle Scholar
  34. Hansen V.L. (1981). Spaces of maps into Eilenberg-Mac Lane spaces. Canad. J. Math. 33(4): 782–785 MATHMathSciNetGoogle Scholar
  35. Hatcher A. (2002). Algebraic Topology. Cambridge University Press, Cambridge MATHGoogle Scholar
  36. Jost J. and Yau S.-T. (1986). The strong rigidity of locally symmetric complex manifolds of rank one and finite volume. Math. Ann. 275(2): 291–304 MATHCrossRefMathSciNetGoogle Scholar
  37. Kapovich, M.: Hyperbolic manifolds and discrete groups. In: Progress in Mathematics, vol. 183. Birkhäuser Boston Inc., Boston, MA (2001)Google Scholar
  38. Korevaar N.J. and Schoen R.M. (1997). Global existence theorems for harmonic maps to non-locally compact spaces. Comm. Anal. Geom. 5(2): 333–387 MATHMathSciNetGoogle Scholar
  39. Kuessner, T.: Multicomplexes, bounded cohomology and additivity of simplicial volume. arXiv:math.AT/0109057Google Scholar
  40. Long D.D. and Reid A.W. (2000). On the geometric boundaries of hyperbolic 4-manifolds. Geom. Topol. 4: 171–178 MATHCrossRefMathSciNetGoogle Scholar
  41. Long D.D. and Reid A.W. (2002). All flat manifolds are cusps of hyperbolic orbifolds. Algebr. Geom. Topol. 2: 285–296 MATHCrossRefMathSciNetGoogle Scholar
  42. McReynolds D.B. (2004). Peripheral separability and cusps of arithmetic hyperbolic orbifolds. Algebr. Geom. Topol. 4: 721–755 MATHCrossRefMathSciNetGoogle Scholar
  43. Mineyev, I., Yaman, A.: Relative Hyperbolicity and Bounded Cohomology. preprint (2006), www.math.uiuc.edu/mineyev/. Accessed 3 Jan 2007
  44. Mostow G.D. and Siu Y.-T. (1980). A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math. (2) 112(2): 321–360 CrossRefMathSciNetGoogle Scholar
  45. Osin, D.V.: Small cancellations over relatively hyperbolic groups and embedding theorems. arXiv:math.GR/0411039Google Scholar
  46. Osin D.V. (2006). Relatively hyperbolic groups: intrinsic geometry, algebraic properties and algorithmic problems, Mem. Amer. Math. Soc. 179(843): vi+100 MathSciNetGoogle Scholar
  47. Paulin, F.: Outer Automorphisms of Hyperbolic Groups and Small Actions on R-trees. In: Roger C. Alperin (ed.) Arboreal group theory (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ., vol. 19, pp. 331–343. Springer, New York (1991)Google Scholar
  48. Rips E. and Sela Z. (1994). Structure and rigidity in hyperbolic groups. I. Geom. Funct. Anal. 4(3): 337–371 MATHCrossRefMathSciNetGoogle Scholar
  49. Spanier, E.H.: Algebraic Topology, Springer-Verlag, New York (1981), Corrected reprintGoogle Scholar
  50. Sun X. (2003). Regularity of harmonic maps to trees. Amer. J. Math. 125(4): 737–771 MATHCrossRefMathSciNetGoogle Scholar
  51. Swenson, E.L.: Quasi-convex groups of isometries of negatively curved spaces, Topology Appl. 110(1), 119–129 (2001). Geometric topology and geometric group theory. Milwaukee, WI (1997)Google Scholar
  52. Szczepański A. (2002). Examples of relatively hyperbolic groups. Geom. Dedicata 93: 139–142 CrossRefMathSciNetMATHGoogle Scholar
  53. Thurston, W.P.: The Geometry and Topology of Three-manifolds, version 2002. http://www.msri.org/publications/books/gt3m. Accessed 3 Jan 2007
  54. Tukia P. (1994). Convergence groups and Gromov’s metric hyperbolic spaces. New Zealand J. Math. 23(2): 157–187 MATHMathSciNetGoogle Scholar
  55. Whitehead, G.W.: Elements of homotopy theory. In: Graduate Texts in Mathematics, vol 61. Springer-Verlag, New York (1978)Google Scholar
  56. Yamaguchi T. (1997). Simplicial volumes of Alexandrov spaces. Kyushu J. Math. 51(2): 273–296 MATHCrossRefMathSciNetGoogle Scholar
  57. Yaman A. (2004). A topological characterisation of relatively hyperbolic groups. J. Reine Angew. Math. 566: 41–89 MATHMathSciNetGoogle Scholar
  58. Yeung S.-K. (1991). Compactification of Kähler manifolds with negative Ricci curvature. Invent. Math. 106(1): 13–25 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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