#### Volume 1, issue 1 (2001)

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On asymptotic dimension of groups

### G Bell and A Dranishnikov

Algebraic & Geometric Topology 1 (2001) 57–71
 arXiv: math.GR/0012006
##### 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: $asdimA{\ast }_{C}B<\infty$.

B) Suppose that ${G}^{\prime }$ is an HNN extension of a group $G$ with $asdimG<\infty$. Then $asdim{G}^{\prime }<\infty$.

C) Suppose that $\Gamma$ is Davis’ group constructed from a group $\pi$ with $asdim\pi <\infty$. Then $asdim\Gamma <\infty$.

##### Keywords
Asymptotic dimension, amalgamated product, HNN extension
##### Mathematical Subject Classification 2000
Primary: 20H15
Secondary: 20E34, 20F69