Abstract
The main result of this paper is that any 3-dimensional manifold with a finite group action is equivariantly invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology 3-sphere also acts on a hyperbolic homology 3-sphere. The theorem has other corollaries, including the existence of infinitely many hyperbolic homology spheres that support free Zp-actions that do not extend over any contractible manifolds, and (from the non-equivariant version of the theorem) infinitely many that bound homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on 3-manifolds is antisymmetric, and thus a partial order.