The symmetric groups, the general linear groups, and the automorphism groups of free groups are examples of families of groups that arise as symmetry groups of algebraic structures but that are also dear to topologists. In this talk I will put their study in a common context that also encompasses many other examples of interest. For instance, in joint work with Nathalie Wahl, this point of view led to a computation of the homology of the Higman-Thompson groups. It is also key to progress on groups related to braid groups and to mapping class groups of universal nil-manifolds that I would like to present.