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Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l_i$ for $i \in \{1,2,3\}$. For any two distinct boundary components, is the length of the shortest geodesic connecting them already determined? If so, is there a simple formula for these lengths as a function of the $l_i$?

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Yes. Cutting along the shortest geodesics $\alpha, \beta, \gamma$ between the pants boundaries produces two congruent right angled hexagons. They are congruent because a right angled hexagon is determined by a triple of sides (in this case, $\alpha, \beta, \gamma).$ This means that each of the hexagons has sides $\alpha, l_1/2, \beta, l_2/2, \gamma, l_3/2.$ Now, the odd sides of a right angled hexagon can be determined from the even sides. See, for example, Fenchel's Elementary Geometry in Hyperbolic Space, available on Google Books, but in any case, the relevant formula (for the hexagon $a, \beta, c, \alpha, b, \gamma$) is:

$$\cosh c = \sinh a \sinh b \cosh \gamma - \cosh a \cosh b.$$

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