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Quasi-isometry and Rigidity

  • Parameswaran SankaranEmail author
Conference paper
  • 38 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 302)

Abstract

This is a brief exposition on the quasi-isometric rigidity of irreducible lattices in Lie groups. The basic notions in coarse geometry are recalled and illustrated. It is beyond the scope of these notes to go into the proofs of most of the results stated here. We shall be content with pointing the reader to standard references for detailed proofs. These notes are based on my talk in the International Conference on Mathematics and its Analysis and Applications in Mathematical Modelling held at Jadavpur University, Kolkata, in December 2017.

Keywords

Quasi-isometry Invariants Rigidity 

Mathematics Subject classification:

20F32 20E99 

References

  1. 1.
    Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)Google Scholar
  2. 2.
    Bridson, M.R., Gersten, S.M.: The optimal isoperimetric inequality for torus bundles over the circle. Quart. J. Math. Oxford Ser. 47(2)(185), 1–23 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Eskin, A.: Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces. J. Am. Math. Soc. 11, 321–361 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eskin, A., Farb, B.: Quasi-flats and rigidity in higher rank symmetric spaces. J. Am. Math. Soc. 10, 653–692 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Farb, B.: The quasi-isometry classification of lattices in semisimple Lie groups. Math. Res. Lett. 4, 705–717 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Farb, B., Schwartz, R.: The large-scale geometry of Hilbert modular groups. J. Diff. Geom. 44, 435–478 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Druţu, C., Kapovich, M.: Geometric group theory. with an appendix by Bogdan Nica. In: American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, vol. 63 (2018)Google Scholar
  8. 8.
    Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53, 53–73 (1981)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gromov, M., Pansu, P.: Rigidity of lattices: an introduction. In: Geometric Topology: recent Developments (Montecatini Terme, 1990), Lecture Notes in Math, vol. 1504, pp. 39–137. Springer, Berlin (1990)Google Scholar
  10. 10.
    Kleiner, B., Leeb, B.: Rigidity of quasi-isometries for symmetric spaces for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math. 86, 115–197 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Koranyi, A., Reimann, H.M.: Foundations for the theory of quasiconformal mappings on the Heisenberg group. Adv. Math. 111, 1–87 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mostow, G.D.: Quasiconformal mappings in n-space and rigidity of hyperbolic space forms. Publ. Inst. Hautes Études Sci. 34, 53–104 (1968)CrossRefGoogle Scholar
  13. 13.
    Pansu, P.: Metriques de Carnot-Caratheodory et quasiisometries des espaces symmetriques de rang un. Ann. Math. 129, 1–60 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Raghunathan, M.S.: Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, vol. 68, Springer, Berlin (1972)CrossRefGoogle Scholar
  15. 15.
    Sankaran, P.: On homeomorphisms and quasi-isometries of the real line. Proc. Am. Math. Soc. 134(7), 1875–1880 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Schwartz, R.: The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math. 82(1995), 133–168 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Schwartz, R.: Quasi-isometric rigidity and diophantine approximation. Acta Math. 177, 75–112 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tukia, P.: Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group. Acta Math. 154, 153–193 (1985)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zimmer, R.J.: Ergodic Theory and Semisimple Lie Groups. Birkhaüser, Boston (1984)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Chennai Mathematical Institute, H1 SIPCOT IT ParkChennaiIndia
  2. 2.Institute of Mathematical Sciences, (HBNI)ChennaiIndia

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