Hyperbolic groups with almost finitely presented subgroups (with an appendix by Robert Kropholler and Federico Vigolo)

In this paper, we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties.We give the first examples of subgroups of hyperbolic groups which are of type FP_2 but not finitely presented. We give uncountably many groups of type FP_2 with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type FP_3 .