A locally hyperbolic 3-manifold that is not hyperbolic

TL;DR

The paper constructs a specific locally hyperbolic 3-manifold with no divisible subgroup in its fundamental group, demonstrating it is not homeomorphic to any complete hyperbolic manifold, thus answering a longstanding question.

Contribution

It provides a counterexample of a locally hyperbolic 3-manifold that is not globally hyperbolic, addressing a question posed by Agol.

Findings

Constructed a locally hyperbolic 3-manifold with no divisible subgroup

Proved this manifold is not homeomorphic to any complete hyperbolic manifold

Answered a question of Agol regarding hyperbolic 3-manifolds

Abstract

We construct a locally hyperbolic 3-manifold $M_\infty$ such that $\pi_ 1(M_\infty)$ has no divisible subgroup. We then show that $M_\infty$ is not homeomorphic to any complete hyperbolic manifold. This answers a question of Agol [DHM06,Mar07].