The geometry of mapping tori of graphs

The well-known subgroup tameness theorem for hyperbolic 3-manifold groups characterises precisely when a finitely generated subgroup is quasi-convex. As a corollary, one can obtain a characterisation of hyperbolic 3-manifold groups that are locally quasi-convex as those that are not virtually {compact surface}-by-cyclic. Although a version of the subgroup tameness theorem for the class of mapping tori of graphs remains a difficult open problem, I will instead show that an analogous characterisation of local quasi-convexity amongst mapping tori of graphs does indeed hold. I will explain the ideas that go into the proof, discuss a generalisation to the relatively hyperbolic setting and mention some applications to one-relator groups.