Abstract
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.
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Arzhantseva, G.N., Guba, V.S., Sapir, M.V.: Metrics on diagram groups and uniform embeddings in Hilbert space. ArXiv GR/0411605 (2005)
Assouad P.: Plongements lipschitziens dans R n. Bull. Soc. Math. France 111(4), 429–448 (1983)
Bourgain J.: The metrical interpretation of superreflexivity in Banach spaces. Israel J. Math. 56(2), 222–230 (1986)
de Cornulier, Y., Tessera, R.: Quasi-isometrically embedded trees. In preparation (2005)
Guentner E., Kaminker J.: Exactness and uniform embeddability of discrete groups. J. Lond. Math. Soc. 70, 703–718 (2004)
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)
Laakso T.J.: Ahlfors Q-regular spaces with arbitrary Q > 1 admiting weak Poincaré inequality. Geom. Funct. Ann. 10(1), 111–123 (2000)
Laakso T.J.: Plane with A ∞-weighted metric not bi-Lischitz embeddable to R n. Bull. Lond. Math. Soc. 34(6), 667–676 (2002)
Lafforgue, V.: Un renforcement de la propriété (T). Preprint (2006)
Lubotzky, A.: Discrete Groups, Expanding Graphs and Invariant Measures, Progress in Mathematics, vol. 125. Birkhauser Verlag (1994)
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)
Roe J.: Warped cones and property A. Geom. Topol. Pub. 9, 163–178 (2005)
Tessera, R.: Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces. Math.GR/0603138 (2006)
Tu J.L.: Remarks on Yu’s “Property A” for discrete metric spaces and groups. Bull. Soc. Math. France 129, 115–139 (2001)
Yu G.: The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139, 201–240 (2000)
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Tessera, R. Quantitative property A, Poincaré inequalities, L p-compression and L p-distortion for metric measure spaces. Geom Dedicata 136, 203–220 (2008). https://doi.org/10.1007/s10711-008-9286-5
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DOI: https://doi.org/10.1007/s10711-008-9286-5
Keywords
- Uniform embeddings of metric spaces into Banach spaces
- Property A
- Poincare inequalities
- Hilbert compression
- Hilbert distortion