Figure eight geodesics on 2-orbifolds (Q260306)

Let \(p,q,r\) be positive integers satisfying \(p^{-1}+q^{-1}+r^{-1}<1\), and let \(T\) be a hyperbolic triangle in the hyperbolic plane \(\mathbf{H}^2\) with interior angles \(\pi/p,\pi/q\), and \(\pi/r\). Let \(\mathcal{O}(p,q,r)\) be the 2-orbifold obtained by taking the quotient \(\mathbf{H}^2/\Gamma\) where \(\Gamma\) is the discrete subgroup of orientation-preserving isometries of \(\mathbf{H}^2\) generated by the reflections in the geodesic sides of \(T\). For a hyperbolic 2-orbifold, the author defines a figure eight geodesic to be a once self-intersecting closed geodesic that does not pass through the singular points. The main result shows that the shortest figure eight geodesic on any triangle group orbifold is the unique one on \(\mathcal{O}(3,3,4)\), which bounds the orbifold points of order three and has length \(2\cosh^{-1}((1+\sqrt{2})/2)\). As a corollary, using a result of W.~Thurston contained in a preprint, the author is able to prove that this same figure eight geodesic is the shortest among all figure eight geodesics on any hyperbolic 2-orbifold without orbifold points of order two.