A locally hyperbolic 3-manifold that is not hyperbolic

We construct a locally hyperbolic 3-manifold $M_{\mathrm{\infty }}$ such that $\pi_ 1(M_\infty)$hasnodivisiblesubgroup\mathrm{.}Wethenshowthat$M_\infty$ is not homeomorphic to any complete hyperbolic manifold. This answers a question of Agol [DHM06,Mar07].