A metric invariant of Möbius transformations

Abstract

The complex unit disk D = {z ∈ C : |z| < 1} is endowed with Möbius addition ⊕M defined by w ⊕M z = w + z 1 + wz. We prove that the metric dT defined on D by dT (w, z) = tan −1 | − w ⊕M z| is an invariant of Möbius transformations carrying D onto itself. We also prove that (D, dT) and (D, dP) , where dP denotes the Poincaré metric, have the same isometry group and then classify the isometries of (D, dT) .

Geometry of gyrogroups via Klein's approach

“…In this section, we focus on the property of transitivity and n-transitivity of gyrogroups. These results are abstract versions of Corollary 6 in [1], Theorem 2.7 in [9], Theorem 2.6 in [8], and Theorem 1 in [3]. In view of Theorem 3.3, it is natural to ask whether the geometry (G, Γ m ) is sharply transitive.…”

“…The study of geometry of abstract gyrogroups in this section is inspired by our three research articles [3,8,9].…”

“…Example 3.12. In the gyrogroup K 16 = {0, 1, 2, 3,4,5,6,7,8,9,10,11,12,13,14 So far we have no example of a gyrogroup G such that (G, Γ m ) is n-transitive for some n ≥ 2. This leads to the following question.…”