The talk is based on joint work with Matt Durham and Yair Minsky where we construct metrics that satisfy a CAT(0)-like inequality involving a sublinear error for a large class of groups including mapping class groups and a lot of their quotients, most 3-manifold groups, extra-large Artin groups, extension of Veech groups and multicurve stabilisers, and many others. These metrics in turn allow one to construct contractible Rips complexes with "nice" compactifications, and to use those to show that the groups in our large class satisfy the Farrell-Jones conjecture.