Large-scale geometry has provided useful tools for understanding manifolds and groups. Guoliang Yu sparked widespread interest in Gromov’s notion of finite asymptotic dimension when he proved the Novikov Conjecture for torsion-free groups G that have finite asymptotic dimension and a cocompact model for the universal space EG. Recently, Guentner, Tessera, and Yu introduced a new concept called finite decomposition complexity (FDC), which they used to significantly generalize Yu's work. The collection of countable groups, considered as metric spaces with a proper left-invariant metric, that have FDC is quite large. Guentner, Tessera, and Yu showed that it contains countable subgroups of GL(n, R), where R is any commutative ring, countable subgroups of almost connected Lie groups, hyperbolic groups, and elementary amenable groups. They also showed that the class of groups with FDC has nice inheritance properties. It is closed under subgroups, extensions, free products with amalgamation, HNN extensions, and countable direct unions. The FDC and weak FDC conditions that they introduced have important topological consequences. In particular, a finitely generated group with weak FDC satisfies the Novikov Conjecture.

In this talk I will present the basic properties of these notions, along with several examples and open problems along the way. I will also discuss recent joint work with Andrew Nicas on finite decomposition complexity.