As part of his celebrated hyperbolization theorem, Thurston proved that a fibered 3-manifold M admits a hyperbolic structure if and only if M is the mapping torus of a pseudo-Anosov homeomorphism. Thinking in terms of the induced splitting of the fundamental group of M, Thurstona**s theorem suggests the possibility of hyperbolizing more general group extensions that live beyond the world of 3-manifold topology.

In this talk I will discuss recent work developing the theory of hyperbolic extensions of free groups and highlight the surprising similarities these findings share with Thurston's original theory. In addition to surveying the geometry of these hyperbolic extensions, I will also present some new results which quantitatively describe how topological and combinatorial data is transferred between various fibrations of the 3-manifold M. This talk includes joint work with S. Dowdall and Y. Minsky.