A locally hyperbolic 3-manifold that is not hyperbolic

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].

11 pages, 5 figures