  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.
