Hyperbolization of infinite-type 3-manifolds

We study the class $\mathcal M^B$ of 3-manifolds $M$ that have a compact exhaustion $M=\cup_{i\in\mathbb N} M_i$ satisfying: each $M_i$ is hyperbolizable with incompressible boundary and each component of $\partial M_i$ has genus at most $g= g(M)$. For manifolds in $\mathcal M^{B}$ we give necessary and sufficient topological conditions that guarantee the existence of a complete hyperbolic metric.