A theorem of Thurston states that the mapping torus of a closed surface with negative Euler characteristic admits a complete hyperbolic metric if and only if the monodromy is pseudo-Anosov. I will talk about the generalizations of the analogous problem for graphs. In particular, we give conditions on characterizing hyperbolicity phenomena in free-by-cyclic groups, and more general free group extensions, and show how the geometry of the extension is reflected in the dynamics of the monodromy. (Joint with Matt Clay).