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Transactions of the American Mathematical Society

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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 | References | Similar Articles | Additional Information

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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  • 1. G. Bell, Asymptotic properties of groups acting on complexes, Proc. Amer. Math. Soc. 133 (2005), no. 2, 387-396. MR 2093059 (2005h:20101)
  • 2. G. Bell and A. Dranishnikov, On asymptotic dimension of groups, Algebr. Geom. Topol. 1 (2001), 57-71. MR 1808331 (2001m:20062)
  • 3. -, On asymptotic dimension of groups acting on trees, Geom. Dedicata 103 (2004), 89-101. MR 2034954 (2005b:20078)
  • 4. G. Bell, A. Dranishnikov, and J. Keesling, On a formula for the asymptotic dimension of free products, Submitted, 2004.
  • 5. M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer, 1999. MR 1744486 (2000k:53038)
  • 6. G. Carlsson and B. Goldfarb, On homological coherence of discrete groups, J. Algebra 276 (2004), 502-514. MR 2058455 (2005a:20078)
  • 7. A. Dranishnikov, Asymptotic topology, Russian Math. Surveys 55 (2000), no. 6, 71-116. MR 1840358 (2002j:55002)
  • 8. -, On asymptotic inductive dimension, JP Jour. Geometry & Topology 1 (2001), no. 3, 239-247. MR 1890851 (2003a:54038)
  • 9. A. Dranishnikov and T. Januszkiewicz, Every Coxeter group acts amenably on a compact space, Topology Proc. 24 (1999), 135-141. MR 1802681 (2001k:20082)
  • 10. A. Dranishnikov and M. Zarichnyi, Universal spaces for asymptotic dimension, Topology Appl. 140 (2004), 203-225. MR 2074917 (2005e:54032)
  • 11. R. Engelking, Theory of dimensions finite and infinite, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, 1995. MR 1363947 (97j:54033)
  • 12. M. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory, London Math. Soc. Lecture Note Ser. (G. Niblo and M. Roller, eds.), no. 182, 1993. MR 1253544 (95m:20041)
  • 13. L. Ji, Asymptotic dimension of arithmetic groups., preprint (2003).
  • 14. J. Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, AMS, 2003. MR 2007488 (2004g:53050)
  • 15. G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Annals of Mathematics 147 (1998), no. 2, 325-355. MR 1626745 (99k:57072)

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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

A. N. Dranishnikov
Affiliation: Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105

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

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