Extension of the Švarc–Milnor lemma to gyrogroups

Abstract

A strongly generated gyrogroup is a gyrogroup that comes with a specific generating set invariant under its gyroautomorphisms, which may be viewed as a suitable generalization of a group. The achievement of constructing a word metric on a strongly generated gyrogroup, together with the recent notion of a gyrogroup action, prompts us to extend the Švarc–Milnor lemma (also known as the fundamental lemma of geometric group theory) to the case of gyrogroups.