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It is known that the shortest non-simple closed geodesic on an orientable hyperbolic 2-orbifold passes through an orbifold point of the orbifold (Nakanishi in Tohoku Math J (2) 41:527–541, 1989). This raises questions about minimal length non-simple closed geodesics disjoint from the orbifold points. Here, we explore once self-intersecting closed geodesics disjoint from the orbifold points of the orbifold, called figure eight geodesics. Using fundamental domains and basic hyperbolic trigonometry, we identify and classify all figure eight geodesics on triangle group orbifolds. This classification allows us to find the shortest figure eight geodesic on a triangle group orbifold, namely the unique one on the (3,3,4)-triangle group orbifold. We then show that this same curve is the shortest figure eight geodesic on a hyperbolic 2-orbifold without orbifold points of order two.
Computational Methods and Function Theory – Springer Journals
Published: Jul 22, 2015
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