A locally hyperbolic 3-manifold that is not hyperbolic

Abstract

We construct a locally hyperbolic 3-manifold M ∞ M_\infty such that π 1 ( M ∞ ) \pi _1(M_\infty ) has no divisible subgroup. We then show that M ∞ M_\infty is not homeomorphic to any complete hyperbolic manifold. This answers a question of Agol.