Equivariant hyperbolization of $3$-manifolds via homology cobordisms

TL;DR

This paper proves that any 3-manifold with a finite group action can be transformed into a hyperbolic manifold via equivariant homology cobordism, leading to new insights into group actions on 3-spheres and hyperbolic structures.

Contribution

It introduces a method to equivariantly hyperbolize 3-manifolds using homology cobordisms, extending hyperbolic structures to manifolds with group actions.

Findings

Existence of hyperbolic equivariant corks for many finite groups

Any finite group action on a homology 3-sphere extends to a hyperbolic homology 3-sphere

Infinite families of hyperbolic homology spheres with specific group actions

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 applications, including establishing the existence of an infinite number of hyperbolic homology spheres with a free $Z_p$ action that does not extend to any contractible manifold. A non-equivariant version yields an infinite number of hyperbolic integer homology spheres that bound integer homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on $3$-manifolds is antisymmetric.