Geometry of gyrogroups via Klein's approach

TL;DR

This paper explores the geometric structure of gyrogroups using Klein's approach, analyzing their symmetry properties and invariant sets through algebraic and transformation group techniques.

Contribution

It introduces a geometric framework for gyrogroups via Klein's approach, focusing on transitivity and invariance properties of their transformation groups.

Findings

Characterization of coset spaces as minimally invariant sets

Open balls of equal radius form minimally invariant sets in normed gyrogroups

Analysis of n-transitivity in gyrogroup geometries

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 $(G, \Gamma_m)$ for any normed gyrogroup $G$, where $\Gamma_m$ is a suitable group of isometries of $G$.