Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T11:23:13.596Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform Embeddings into Hilbert Space and a Question of Gromov

Published online by Cambridge University Press:  20 November 2018

A. N. Dranishnikov ,
G. Gong ,
V. Lafforgue  and
G. Yu
A. N. Dranishnikov
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA, email: [email protected]
G. Gong
Affiliation:
Department of Mathematics, University of Puerto Rico, Rio Piedras, San Juan, PR 00931 USA, email: [email protected]
V. Lafforgue
Affiliation:
Laboratoire de Mathématiques de l’Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France, email: [email protected]
G. Yu
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309–0395, USA, email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Gromov introduced the concept of uniform embedding into Hilbert space and asked if every separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov’s question negatively.

Keywords

46C05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 1 , 01 March 2002 , pp. 60 - 70
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Bekka, M. E. B., Cherix, P. A. and Valette, A., Proper affine isometric actions of amenable groups. In: Novikov Conjectures, Index Theorems and Rigidity, Vol. 2 (eds. S. Ferry, A. Ranicki and J. Rosenberg), Cambridge University Press 1995, 1–4.Google Scholar
[2] Connes, A., Gromov, M. and Moscovici, H., Group cohomology with Lipschitz control and higher signatures. Geom. Funct. Anal. 3 (1993), 1–78.Google Scholar
[3] Dranishnikov, A. N., On generalized amenability. Preprint IHES/M/99/61, 1999.Google Scholar
[4] Dranishnikov, A. N. and Januszkiewicz, T., On Higson-Roe amenability of Coxeter groups. Topology Proceedings, to appear.Google Scholar
[5] Enflo, P., On a problem of Smirnov. Ark. Mat. 8 (1969), 107–109.Google Scholar
[6] Gong, G. and Yu, G., Volume growth and positive scalar curvature. Preprint, 1998.Google Scholar
[7] Gromov, M., Asymptotic invariants for infinite groups. In: Geometric Group Theory (eds. G. A. Niblo and M. A. Roller), Cambridge University Press, 1993, 1–295.Google Scholar
[8] Gromov, M., Problems (4) and (5). In: Novikov Conjectures, Index Theorems and Rigidity, Vol. 1 (eds. S. Ferry, A. Ranicki and J. Rosenberg), Cambridge University Press, 1995, 67.Google Scholar
[9] Gromov, M., Positive curvature, macroscopic dimension, spectral gaps and higher signatures. Functional Analysis on the eve of the 21st century, Vol. 2, Progr. Math. 132 (1996), 1–213.Google Scholar
[10] Higson, N. and Kasparov, G. G., Operator K-theory for groups which act properly and isometrically on Hilbert space. Electron. Res. Announc. Amer.Math. Soc. 3 (1997), 131–141.Google Scholar
[11] Higson, N. and Roe, J., Amenable group actions and the Novikov conjecture. Preprint, 1998.Google Scholar
[12] Manin, Y. I., A Course in Mathematical Logic. Springer-Verlag, 1977.Google Scholar
[13] Roe, J., Index Theory, Coarse Geometry, and Topology of Manifolds. CBMS Regional Conf. Series in Math. 90, Amer.Math. Soc., 1996.Google Scholar
[14] Sela, Z., Uniform embeddings of hyperbolic groups in Hilbert spaces. Israel J. Math. 80 (1992), 171–181.Google Scholar
[15] Schoenberg, I. J., Remarks to Maurice Fréchet's article. Ann.Math. 36 (1935), 724–732.Google Scholar
[16] Valiev, M. K., Examples of universal finitely presented groups. Dokl. Akad. Nauk. SSSR 211 (1973), 265–268.Google Scholar
[17] Yu, G., The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent.Math. (1) 139 (2000), 201–240.Google Scholar