Algebraic and Geometric Topology 1 (2001), paper no. 4, pages 57-71.

On asymptotic dimension of groups

G. Bell and A. Dranishnikov

Abstract. We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems.
A) An amalgamated product of asymptotically finite dimensional groups has finite asymptotic dimension: asdim A *_C B < infinity.
B) Suppose that G' is an HNN extension of a group G with asdim G < infinity. Then asdim G'< infinity.
C) Suppose that \Gamma is Davis' group constructed from a group \pi with asdim\pi < infinity. Then asdim\Gamma < infinity.

Keywords. Asymptotic dimension, amalgamated product, HNN extension

AMS subject classification. Primary: 20H15. Secondary: 20E34, 20F69.

DOI: 10.2140/agt.2001.1.57

E-print: arXiv:math.GR/0012006

Submitted: 11 December 2000. (Revised: 12 January 2001.) Accepted: 12 January 2001. Published: 27 January 2001.

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G. Bell and A. Dranishnikov
University of Florida, Department of Mathematics,
PO Box 118105, 358 Little Hall,
Gainesville, FL 32611-8105, USA

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