Constructions preserving Hilbert space uniform embeddability of discrete groups

Dadarlat, Marius; Guentner, Erik

doi:10.1090/S0002-9947-03-03284-7

Constructions preserving Hilbert space uniform embeddability of discrete groups
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by Marius Dadarlat and Erik Guentner PDF

Trans. Amer. Math. Soc. 355 (2003), 3253-3275 Request permission

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

Juan M. Alonso and Martin R. Bridson, Semihyperbolic groups, Proc. London Math. Soc. (3) 70 (1995), no. 1, 56–114. MR 1300841, DOI 10.1112/plms/s3-70.1.56
C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their $C^*$-algebras, Trans. Amer. Math. Soc. 354 (2002), 4153–4178.
C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 36, L’Enseignement Mathématique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain. MR 1799683
Gilbert Baumslag, Topics in combinatorial group theory, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR 1243634, DOI 10.1007/978-3-0348-8587-4
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.
Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette, Groups with the Haagerup property, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001. Gromov’s a-T-menability. MR 1852148, DOI 10.1007/978-3-0348-8237-8
A. N. Dranishnikov, G. Gong, V. Lafforgue, and G. Yu, Uniform embeddings into Hilbert space and a question of Gromov, Canad. Math. Bull. 45 (2002), no. 1, 60–70. MR 1884134, DOI 10.4153/CMB-2002-006-9
K. Dykema, Exactness of reduced amalgamated free product $C^*$-algebras, Preprint, 1999.
Kenneth J. Dykema, Free products of exact groups, $C^*$-algebras (Münster, 1999) Springer, Berlin, 2000, pp. 61–70. MR 1796910
Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
S. M. Gersten, Bounded cocycles and combings of groups, Internat. J. Algebra Comput. 2 (1992), no. 3, 307–326. MR 1189238, DOI 10.1142/S0218196792000190
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
Erik Guentner and Jerome Kaminker, Exactness and the Novikov conjecture, Topology 41 (2002), no. 2, 411–418. MR 1876896, DOI 10.1016/S0040-9383(00)00036-7
Eberhard Kirchberg and Simon Wassermann, Exact groups and continuous bundles of $C^*$-algebras, Math. Ann. 315 (1999), no. 2, 169–203. MR 1721796, DOI 10.1007/s002080050364
Eberhard Kirchberg and Simon Wassermann, Permanence properties of $C^*$-exact groups, Doc. Math. 4 (1999), 513–558. MR 1725812
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, DOI 10.1016/S0764-4442(00)00248-2
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
G. Skandalis, J. L. Tu, and G. Yu, The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), no. 4, 807–834. MR 1905840, DOI 10.1016/S0040-9383(01)00004-0
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, DOI 10.24033/bsmf.2388
P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294