Asymptotically spherical groups

Abstract

We define a notion of asymptotically spherical topological groups, which says that spheres of large radius with respect to any maximal length function are still spherical with respect to any other maximal length function. This is a strengthening of a related condition introduced by Sebastian Hurtado, which we call bounded eccentricity. Our main result is a partial characterization of which groups are asymptotically spherical, and we also give an example of a discrete, bounded eccentric group who fails to be asymptotically spherical.