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: .
B) Suppose that is an HNN extension of a group with . Then .
C) Suppose that is Davis’ group constructed from a group with . Then .
"On asymptotic dimension of groups." Algebr. Geom. Topol. 1 (1) 57 - 71, 2001. https://doi.org/10.2140/agt.2001.1.57