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Uniform embeddings of bounded geometry spaces into reflexive Banach space

Authors: Nathanial Brown and Erik Guentner
Journal: Proc. Amer. Math. Soc. 133 (2005), 2045-2050
MSC (2000): Primary 46B07
Published electronically: January 21, 2005
MathSciNet review: 2137870
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Abstract: We show that every metric space with bounded geometry uniformly embeds into a direct sum of $l^p ({\mathbb N})$ spaces ($p$'s going off to infinity). In particular, every sequence of expanding graphs uniformly embeds into such a reflexive Banach space even though no such sequence uniformly embeds into a fixed $l^p ({\mathbb N})$ space. In the case of discrete groups we prove the analogue of a-$T$-menability - the existence of a metrically proper affine isometric action on a direct sum of $l^p ({\mathbb N})$ spaces.

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  • 1. M. E. B. Bekka, P.-A. Cherix, and A. Valette, Proper affine isometric actions of amenable groups, Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, Cambridge, 1995, pp. 1–4. MR 1388307, 10.1017/CBO9780511629365.003
  • 2. Steven C. Ferry, Andrew Ranicki, and Jonathan Rosenberg (eds.), Novikov conjectures, index theorems and rigidity. Vol. 2, London Mathematical Society Lecture Note Series, vol. 227, Cambridge University Press, Cambridge, 1995. Including papers from the conference held at the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, September 6–10, 1993. MR 1388306
  • 3. M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
  • 4. Misha Gromov, Spaces and questions, Geom. Funct. Anal. Special Volume (2000), 118–161. GAFA 2000 (Tel Aviv, 1999). MR 1826251
  • 5. M. Gromov. Random walk in random groups. IHES preprint, 2002.
  • 6. E. Guentner, N. Higson, and S. Weinberger. The Novikov conjecture for linear groups. Preprint, 2003.
  • 7. Erik Guentner and Jerome Kaminker, Exactness and the Novikov conjecture, Topology 41 (2002), no. 2, 411–418. MR 1876896, 10.1016/S0040-9383(00)00036-7
  • 8. N. Higson and E. Guentner. $K$-theory of group $C^*$-algebras. In S. Doplicher and R. Longo, editors, Noncommutative Geometry, Lecture Notes in Mathematics, pages 253-262. Springer Verlag, Berlin, 2003.
  • 9. G. Kasparov and G. Yu The coarse geometric Novikov conjecture and uniform convexity. Preprint.
  • 10. Jiří Matoušek, On embedding expanders into 𝑙_{𝑝} spaces, Israel J. Math. 102 (1997), 189–197. MR 1489105, 10.1007/BF02773799
  • 11. N. Ozawa. A note on non-amenability of $B(\ell_p)$ for $p=1,2$. Preprint.
  • 12. Narutaka Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 8, 691–695 (English, with English and French summaries). MR 1763912, 10.1016/S0764-4442(00)00248-2
  • 13. G. Skandalis, J. L. Tu, and G. Yu, The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), no. 4, 807–834. MR 1905840, 10.1016/S0040-9383(01)00004-0
  • 14. Jean-Louis Tu, Remarks on Yu’s “property A” for discrete metric spaces and groups, Bull. Soc. Math. France 129 (2001), no. 1, 115–139 (English, with English and French summaries). MR 1871980
  • 15. Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR 1728880, 10.1007/s002229900032

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Additional Information

Nathanial Brown
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Erik Guentner
Affiliation: Department of Mathematics, 2565 McCarthy Mall, University of Hawaii, Mānoa, Honolulu, Hawaii 96822-2273

Received by editor(s): September 17, 2003
Received by editor(s) in revised form: March 5, 2004
Published electronically: January 21, 2005
Additional Notes: The authors were partially supported by grants from the U.S. National Science Foundation
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.