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Geometriae Dedicata

, Volume 124, Issue 1, pp 143–152 | Cite as

Rigidity of hyperbolic P-manifolds: a survey

  • Jean-François LafontEmail author
Original Paper
  • 32 Downloads

Abstract

In this survey paper, we outline the proofs of rigidity results (Mostow type, quasi-isometric, and Diagram rigidity) for simple, thick, hyperbolic P-manifolds. These are stratified spaces, and are in some sense the simplest non-manifold locally CAT(−1) spaces one can create, having codimension one singularities along embedded totally geodesic submanifolds. All the proofs depend on the highly non-homogenous structure of the boundary at infinity of the (universal covers of the) spaces in question, and in particular, on an understanding of the local topology of the boundary at infinity. We emphasize the similarities and differences in the proofs of the various rigidity results. These results should be viewed as a first step towards understanding stratified locally CAT(−1) spaces.

Keywords

Mostow rigidity Quasi-isometry Diagram of groups Singular spaces CAT(−1) Hyperbolic groups 

Mathematics Subject Classifications (2000)

Primary 20F65 Secondary 20F67 53C24 57N35 57N45 57N50 57N80 

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References

  1. 1.
    Ancel F.D. (1986). Resolving wild embeddings of codimension-one manifolds in manifolds of dimensions greater than 3, Special volume in honor of R. H. Bing (1914–1986). Topol. Appl. 24: 13–40 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ancel F.D. and Cannon J.W. (1979). The locally flat approximation of cell-like embedding relations. Ann. Math. 109(2): 61–86 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bing R.H. (1957). Approximating surfaces by polyhedral ones. Ann. Math. 65(2): 465–483 MathSciNetGoogle Scholar
  4. 4.
    Bridson M.R. and Haefliger A. (1999). Metric Spaces of Non-Positive Curvature. Springer-Verlag, BerlinzbMATHGoogle Scholar
  5. 5.
    Cannon J.W. (1973). A positional characterization of the (n − 1)-dimensional Sierpiński curve in S n(n ≠ 4). Fund. Math. 79: 107–112 zbMATHMathSciNetGoogle Scholar
  6. 6.
    Farb B. (1997). The quasi-isometry classification of lattices in semisimple Lie groups. Math. Res. Lett. 4: 705–717 zbMATHMathSciNetGoogle Scholar
  7. 7.
    Farrell F.T. and Jones L.E. (1989). A topological analogue of Mostow’s rigidity theorem. J. Amer. Math. Soc. 2: 257–370 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Frigerio R. (2004). Hyperbolic manifolds with geodesic boundary which are determined by their fundamental group. Topol. Appl. 145: 69–81 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Frigerio R. (2006). Commensurability of hyperbolic manifolds with geodesic boundary. Geom. Dedicata 118: 105–131 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lafont J.-F. (2004). Rigidity result for certain 3-dimensional singular spaces and their fundamental groups. Geom. Dedicata 109: 197–219 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lafont J.-F. (2006). Strong Jordan separation and applications to rigidity. J. London Math. Soc. 73: 681–700 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lafont J.-F. (2007). Rigidity of geometric amalgamations of free groups. J. Pure Appl. Algebra 209: 771–780 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lafont, J.-F.: A boundary version of Cartan-Hadamard and applications to rigidity. Preprint available on the ArXiv at http://front.math.ucdavis.edu/math.GT/0606629Google Scholar
  14. 14.
    Mostow G.D. (1973). Strong Rigidity of Locally Symmetric Spaces. Princeton University Press, Princeton, N.J.zbMATHGoogle Scholar
  15. 15.
    Sierpinski W. (1918). Un théorème sur les continus. Tohoku Math. J. 13: 300–303zbMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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