The hyperbolic polygons of type (ϵ, n) and Möbius transformations

Abstract

Abstract An n-sided hyperbolic polygon of type (ϵ, n) is a hyperbolic polygon with ordered interior angles $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ, θ1, θ2, …, θn−2, $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ, where 0 < ϵ < $\begin{array}{} \frac{\pi}{2} \end{array} $ and 0 < θi < π satisfying $$\begin{array}{} \displaystyle \sum_{i = 1}^{n-2} \theta_{i}+\Big(\frac{\pi}{2}+\epsilon\Big)+\Big(\frac{\pi}{2}-\epsilon\Big) \lt (n-2)\pi \end{array} $$ and θi + θi+1 ≠ π (1 ≤ i ≤ n − 3), θ1 + ( $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ) ≠ π, θn−2 + ( $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ) ≠ π. In this paper, we present a new characterization of Möbius transformations by using n-sided hyperbolic polygons of type (ϵ, n). Our proofs are based on a geometric approach.