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We introduce a geometric invariant, called finite decomposition complexity(FDC), to study topological rigidity of manifolds. We prove for instance thatif the fundamental group of a compact aspherical manifold M has FDC, and if Nis homotopy equivalent to M, then M x R^n is homeomorphic to N x R^n, for nlarge enough. This statement is known as the stable Borel conjecture. On theother hand, we show that the class of FDC groups includes all countablesubgroups of GL(n,K), for any field K, all elementary amenable groups, and isclosed under taking subgroups, extensions, free amalgamated products, HNNextensions, and direct unions.

A notion of geometric complexity and its application to topological rigidity