On asymptotic dimension of groups

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 $asdim\Gamma <\infty$.