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Constructions preserving Hilbert space uniform embeddability of discrete groups

Authors: Marius Dadarlat and Erik Guentner
Journal: Trans. Amer. Math. Soc. 355 (2003), 3253-3275
MSC (2000): Primary 46L89, 20F65
Published electronically: April 8, 2003
MathSciNet review: 1974686
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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.

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Additional Information

Marius Dadarlat
Affiliation: Department of Mathematics, Purdue University, 1395 Mathematical Sciences Building, West Lafayette, Indiana 47907-1395

Erik Guentner
Affiliation: Mathematics Department, University of Hawaii, Manoa, 2565 McCarthy Mall, Honolulu, Hawaii 96822

Received by editor(s): July 22, 2002
Received by editor(s) in revised form: December 26, 2002
Published electronically: 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.
Article copyright: © Copyright 2003 American Mathematical Society