Asymptotically spherical groups

TL;DR

This paper introduces the concept of asymptotically spherical groups, strengthening previous notions, and provides a partial classification along with a counterexample of a bounded eccentric group that is not asymptotically spherical.

Contribution

It defines asymptotically spherical groups, compares it with bounded eccentricity, and offers a partial characterization and a counterexample.

Findings

Partial classification of asymptotically spherical groups

Existence of bounded eccentric groups that are not asymptotically spherical

Introduction of a new notion strengthening previous conditions

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.