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: asdim A *_C B < infinity. B) Suppose that G&#39; is an HNN extension of a group G with asdim G < infinity. Then asdim G&#39;< infinity. C) Suppose that Γis Davis&#39; group constructed from a group πwith asdimπ< infinity. Then asdimΓ< infinity.