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On the injectivity radius of hyperbolic polyhedra and rational homology 3-sphers

We define the injectivity radius of a Coxeter polyhedron in H to be half the shortest translation length among hyperbolic/loxodromic elements in the orientation-preserving reflection group. We show that, for finite-volume polyhedra, this number is always less than 2.6339... , and for compact polyhedra it is always less than 2.1225... . We apply this result to a question about the injectivity radius of hyperbolic rational homology 3-spheres.