3-manifold group and finite decomposition complexity (Q971833)

Roughly speaking a metric space has finite decomposition complexity if there is an algorithm to decompose the space into nice pieces in a certain way. In the paper under review mainly finitely generated groups with the word length metric are considered. The main result is the following. Suppose Thurston's hyperbolization conjecture is true and \(G\) is the fundamental group of an orientable compact 3-manifold (possibly with boundary), then \(G\) has finite decomposition complexity. The paper has the following structure. The definition of finite decomposition complexity is given which was intoduced by \textit{E. Guentner, R. Tessera} and \textit{G. Yu} in the preprint entitled [Decomposition complexity and stable Borel conjecture]. Then is proved that fundamental groups of surfaces (also with boundary), fundamental groups of Haken and prime Seifert 3-manifolds have finite decomposition complexity. Finally the main result is proved.