#### 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.

Notes on file formats
G. Bell and A. Dranishnikov

University of Florida, Department of Mathematics,

PO Box 118105, 358 Little Hall,

Gainesville, FL 32611-8105, USA

Email: dranish@math.ufl.edu, bell@math.ufl.edu

AGT home page

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