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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory


Authors: G. C. Bell and A. N. Dranishnikov
Journal: Trans. Amer. Math. Soc. 358 (2006), 4749-4764
MSC (2000): Primary 20F69, 20F65; Secondary 20E08, 20E06
Published electronically: April 17, 2006
MathSciNet review: 2231870
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Abstract: We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.


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Additional Information

G. C. Bell
Affiliation: Department of Mathematics, Penn State University, University Park, Pennsylvania 16802
Address at time of publication: Mathematical Sciences, University of North Carolina at Greensboro, Greensboro, North Carolina 27402
Email: bell@math.psu.edu, gcbell@uncg.edu

A. N. Dranishnikov
Affiliation: Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105
Email: dranish@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04088-8
PII: S 0002-9947(06)04088-8
Keywords: Asymptotic dimension, free products, nilpotent groups
Received by editor(s): July 20, 2004
Published electronically: April 17, 2006
Additional Notes: The second author was partially supported by NSF Grant DMS-0305152
Article copyright: © Copyright 2006 American Mathematical Society