Isometry Groups of Hyperbolic Three-Manifolds which are Cyclic Branched Coverings

Abstract

Let M be a hyperbolic three-manifold which is an n-fold cyclic branched covering of a hyperbolic link L in the three-sphere, or more precisely, of the hyperbolic three-orbifold \(\mathcal{O}_n (L)\) whose underlying topological space is the three-sphere and whose singular set, of branching index n, the link L. We say that M has no hidden symmetries (with respect to the given branched covering) if the isometry group of M is the lift of (a subgroup of) the isometry group of the hyperbolic orbifold \(\mathcal{O}_n (L)\) (which is isomorphic to the symmetry group of the link L). It follows from Thurston's hyperbolic surgery theorem that M has no hidden symmetries if n is sufficiently large. Our main result is an explicit numerical version of this fact: we give a constant, in terms of the volume of the complement of L, such that M has no hidden symmetries for all n larger than this constant; we show by examples that a universal constant working for all hyperbolic knots or links does not exist. We give also some results on the possible orders and the structure of the isometry group of M. Finally, we construct sets of four different π-hyperbolic knots which have the same two-fold branched covering (a hyperbolic three-manifold); it is an interesting question for how many different π-hyperbolic knots (or links) this may happen (in the case of hyperbolic knots, for arbitrarily many).