|
Constructions preserving Hilbert space uniform embeddability of discrete groups
Author(s):
Marius
Dadarlat;
Erik
Guentner
Journal:
Trans. Amer. Math. Soc.
355
(2003),
3253-3275.
MSC (2000):
Primary 46L89, 20F65
Posted:
April 8, 2003
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.
References:
-
- 1.
- J. Alonso and M. Bridson, Semihyperbolic groups, Proc. London Math. Soc. 70 (1995), 56-114. MR 95j:20033
- 2.
- C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their
 -algebras, Trans. Amer. Math. Soc. 354 (2002), 4153-4178. - 3.
- C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, with a foreword by Georges Skandalis and Appendix B by E. Germain, Monographies de L'Enseignement Math., vol. 36, L'Enseignement Math., Geneva, 2000. MR 2001m:22005
- 4.
- G. Baumslag, Topics in combinatorial group theory, ETH Lectures in Mathematics, Birkhäuser-Verlag, Basel, 1993. MR 94j:20034
- 5.
- X. Chen, M. Dadarlat, E. Guentner, and G. Yu, Uniform embeddings into Hilbert space and free products of groups, to appear in J. Funct. Anal.
- 6.
- P. Cherix, M. Cowling, P. Jolissaint, P. Julg, and A. Valette, Groups with the Haagerup property. Gromov's a-T-menability, Progress in Mathematics, vol. 197, Birkhäuser-Verlag, Basel, 2001. MR 2002h:22007
- 7.
- A. Dranishnikov, G. Gong, V. Lafforgue, and G. Yu, Uniform embeddings into Hilbert space and a question of Gromov, Canad. Math. Bull. 45 (2002), 60-70. MR 2003a:57043
- 8.
- K. Dykema, Exactness of reduced amalgamated free product
 -algebras, Preprint, 1999. - 9.
- -, Free products of exact groups,
 -algebras (Münster, 1999) (J. Cuntz and S. Echterhoff, eds.), Springer-Verlag, Berlin, 2000, pp. 61-70. MR 2001m:46133 - 10.
- S. Ferry, A. Ranicki, and J. Rosenberg (eds.), Novikov conjectures, index theorems and rigidity, London Mathematical Society Lecture Notes, nos. 226, 227, Cambridge University Press, 1995. MR 96m:57002; MR 96m:57003
- 11.
- S. Gersten, Bounded cocycles and combings of groups, unpublished manuscript, version 5.5, 1991, cf. Internat. J. Algebra Comput. 2 (1992), 307-326. MR 93i:20029
- 12.
- M. Gromov, Asymptotic invariants of infinite groups, London Mathematical Society Lecture Notes, no. 182, pp. 1-295, Cambridge University Press, Cambridge, 1993. MR 95m:20041
- 13.
- E. Guentner and J. Kaminker, Exactness and the Novikov conjecture, Topology 41 (2002), no. 2, 411-418. MR 2003e:46097a
- 14.
- E. Kirchberg and S. Wassermann, Exact groups and continuous bundles of
 -algebras, Mathematische Annalen 315 (1999), 169-203. MR 2000i:46050 - 15.
- -, Permanence properties of
 -exact groups, Documenta Mathematica 4 (1999), 513-558 (electronic). MR 2001i:46089 - 16.
- N. Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 8, 691-695. MR 2001g:22007
- 17.
- J. P. Serre, Trees, Springer-Verlag, New York, 1980, Translation from French of ``Arbres, Amalgames,
 '', Astérisque no. 46. MR 82c:20083 - 18.
- G. Skandalis, J. L. Tu, and G. Yu, The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), 807-834. MR 2003c:58020
- 19.
- J. L. Tu, Remarks on Yu's Property A for discrete metric spaces and groups, Bull. Soc. Math. France 129 (2001), 115-139. MR 2002j:58038
- 20.
- G. Yu, The Coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Inventiones Math. 139 (2000), 201-240. MR 2000j:190057
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
46L89, 20F65
Retrieve articles in all Journals with MSC
(2000):
46L89, 20F65
Additional Information:
Marius
Dadarlat
Affiliation:
Department of Mathematics, Purdue University, 1395 Mathematical Sciences Building, West Lafayette, Indiana 47907-1395
Email:
mdd@math.purdue.edu
Erik
Guentner
Affiliation:
Mathematics Department, University of Hawaii, Manoa, 2565 McCarthy Mall, Honolulu, Hawaii 96822
Email:
erik@math.hawaii.edu
DOI:
10.1090/S0002-9947-03-03284-7
PII:
S 0002-9947(03)03284-7
Received by editor(s):
July 22, 2002
Received by editor(s) in revised form:
December 26, 2002
Posted:
April 8, 2003
Additional Notes:
The first author was supported in part by an MSRI Research Professorship and NSF Grant DMS-9970223. The second author was supported in part by an MSRI Postdoctoral Fellowship and NSF Grant DMS-0071402.
Copyright of article:
Copyright
2003,
American Mathematical Society
|
Comments: webmaster@ams.org