Motivated by discoveries about projection lattices of finite operator algebras, in the 1930s John von Neumann defined and studied a continuous analogue of projective geometries: continuous geometries. Among other remarkable results, von Neumann proved that every continuous geometry (of order at least four) can be coordinatized by some (up to isomorphism unique) ring, and that the continuous rings (i.e., rings corresponding to continuous geometries via this coordinatization theorem) are precisely those irreducible, regular rings which admit a complete rank function. The (necessarily unique) complete rank function of a continuous ring gives rise to a compatible complete metric and thus furnishes the ring with a natural topology. Unit groups of such continuous rings, equipped with the induced rank topology, constitute an interesting family of large topological groups. In the talk, I will report on recent results concerning (extreme) amenability and related properties of topological unit groups of continuous rings. One of the main tools in this study is a new concentration result for convolution products of invariant means.