Geometriae Dedicata

, Volume 136, Issue 1, pp 203–220 | Cite as

Quantitative property A, Poincaré inequalities, L p -compression and L p -distortion for metric measure spaces

  • Romain TesseraEmail author
Original Paper


We introduce a quantitative version of Property A in order to estimate the L p -compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L p -distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincaré inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.


Uniform embeddings of metric spaces into Banach spaces Property A Poincare inequalities Hilbert compression Hilbert distortion 

Mathematics Subject Classification (2000)

51F99 43A85 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arzhantseva, G.N., Guba, V.S., Sapir, M.V.: Metrics on diagram groups and uniform embeddings in Hilbert space. ArXiv GR/0411605 (2005)Google Scholar
  2. 2.
    Assouad P.: Plongements lipschitziens dans R n. Bull. Soc. Math. France 111(4), 429–448 (1983)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bourgain J.: The metrical interpretation of superreflexivity in Banach spaces. Israel J. Math. 56(2), 222–230 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    de Cornulier, Y., Tessera, R.: Quasi-isometrically embedded trees. In preparation (2005)Google Scholar
  5. 5.
    Guentner E., Kaminker J.: Exactness and uniform embeddability of discrete groups. J. Lond. Math. Soc. 70, 703–718 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded geometry, fractals, and low-distortion embeddings. In: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science (2003)Google Scholar
  7. 7.
    Laakso T.J.: Ahlfors Q-regular spaces with arbitrary Q > 1 admiting weak Poincaré inequality. Geom. Funct. Ann. 10(1), 111–123 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Laakso T.J.: Plane with A -weighted metric not bi-Lischitz embeddable to R n. Bull. Lond. Math. Soc. 34(6), 667–676 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lafforgue, V.: Un renforcement de la propriété (T). Preprint (2006)Google Scholar
  10. 10.
    Lubotzky, A.: Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Mathematics, vol. 125. Birkhauser Verlag (1994)Google Scholar
  11. 11.
    Linial, N., Magen, A., Naor, A.: Girth and Euclidean distortion. In: Proc. of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, pp. 705–711 (2002)Google Scholar
  12. 12.
    Roe J.: Warped cones and property A. Geom. Topol. Pub. 9, 163–178 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Tessera, R.: Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces. Math.GR/0603138 (2006)Google Scholar
  14. 14.
    Tu J.L.: Remarks on Yu’s “Property A” for discrete metric spaces and groups. Bull. Soc. Math. France 129, 115–139 (2001)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Yu G.: The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139, 201–240 (2000)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations