Advertisement

Indian Journal of Pure and Applied Mathematics

, Volume 50, Issue 4, pp 877–881 | Cite as

On the center of the group of quasi-isometries of the real line

  • Prateep ChakrabortyEmail author
Article
  • 33 Downloads

Abstract

Let QI(ℝ) denote the group of all quasi-isometries f : ℝ → ℝ. Let Q+(and Q) denote the subgroup of QI(ℝ) consisting of elements which are identity near −∞ (resp. +∞). We denote by QI+(ℝ) the index 2 subgroup of QI(ℝ) that fixes the ends +∞, −∞. We show that QI+(ℝ) ≅ Q+ × Q. Using this we show that the center of the group QI(ℝ) is trivial.

Key words

PL-homeomorphisms quasi-isometry center of group 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgement

The author thanks Aniruddha C. Naolekar, Parameswaran Sankaran, Ajay Singh Thakur and the anonymous referee for their valuable suggestions and comments.

References

  1. 1.
    M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grund. Math. Wiss., 319 (1999), Springer-Verlag, Berlin.Google Scholar
  2. 2.
    M. Brin and C. C. Squier, Groups of piecewise linear homeomorphisms of the real line, Invent. Math., 79 (1985), 485–498.MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math.42 (1996), 215–256.MathSciNetzbMATHGoogle Scholar
  4. 4.
    D. B. A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Math., 22 (1970), 165–173.MathSciNetzbMATHGoogle Scholar
  5. 5.
    M. Gromov and P. Pansu, Rigidity of lattices: An introduction, In: Geometry and Topology: Recent Developments, Eds. P. de Bartolomeis and F. TricerriLect, Notes Math., 1504, Springer-Verlag, Berlin, 1991.Google Scholar
  6. 6.
    J. Grabowski, Free subgroups of diffeomorphism groups, Fund. Math., 131 (1988), 103–121.MathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Sankaran, On homeomorphisms and quasi-isometries of the real line, Proc. of the Amer. Math. Soc., 134 (2005), 1875–1880.MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian National Science Academy 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

Personalised recommendations