Marco Linton

Characterising local quasi-convexity

The well-known subgroup tameness theorem for hyperbolic 3-manifold groups characterises precisely when a finitely generated subgroup is quasi-convex. An immediate consequence is a characterisation of hyperbolic 3-manifold groups that are locally quasi-convex as those that do not contain {compact surface}-by-cyclic subgroups. Although a version of the subgroup tameness theorem for the class of free-by-cyclic groups remains a difficult open problem, I will instead show that an analogous characterisation of local quasi-convexity amongst free-by-cyclic groups does indeed hold. I will also discuss a generalisation to the relatively hyperbolic setting and mention some applications to cubulated groups and to one-relator groups.