Geometry of gyrogroups via Klein's approach

Abstract

Using Klein's approach, geometry can be studied in terms of a space of points and a group of transformations of that space. This allows us to apply algebraic tools in studying geometry of mathematical structures. In this article, we follow Klein's approach to study the geometry $(G, \mathcal{T})$, where $G$ is an abstract gyrogroup and $\mathcal{T}$ is an appropriate group of transformations containing all gyroautomorphisms of $G$. We focus on $n$-transitivity of gyrogroups and also give a few characterizations of coset spaces to be minimally invariant sets. We then prove that the collection of open balls of equal radius is a minimally invariant set of the geometry $(G, Γ_m)$ for any normed gyrogroup $G$, where $Γ_m$ is a suitable group of isometries of $G$.