 Elliptic stabilizer obstruction to a hyperbolic manifold quotient

A nontrivial finite-order elliptic Möbius transformation fixes an axis in hyperbolic space. The quotient metric has cone angle $2\pi /m$ along that axis for a stabilizer of order $m$, so it is locally an orbifold metric rather than a smooth hyperbolic-manifold metric. The underlying topological quotient may nevertheless be a manifold.