## Equivariant hyperbolization and applications

Given a closed 3-manifold Y with the action of a finite group G, we show how to find a closely related hyperbolic G-manifold Y’. The two manifolds are related by an invertible equivariant homology cobordism; a cobordism from Y to Y’ is invertible if there is a second cobordism from Y’ to Y such that the union of the two along Y’ is a product cobordism. I will give a collection of applications in 3 and 4-dimensional topology. (Joint work with Dave Auckly, Hee Jung Kim, and Paul Melvin.)