Discrete groups with finite decomposition complexity

Abstract

In [GTY] we introduced a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. In that article we proved the stable Borel conjecture for a closed aspherical manifold whose universal cover, or equivalently whose fundamental group, has FDC. In this note we continue our study of FDC, focusing on permanence and the relation to other coarse geometric properties. In particular, we prove that the class of FDC groups is closed under taking subgroups, extensions, free amalgamated products, HNN extensions, and direct unions. As consequences we obtain further examples of FDC groups – all elementary amenable groups and all countable subgroups of almost connected Lie groups have FDC.