A hyperbolic group with a finitely presented subgroup that is not of   type $FP_3$
  Brady proved that there are hyperbolic groups with finitely presented subgroups that are not of type $FP_3$ (and hence not hyperbolic). We reprove Brady's theorem by presenting a new construction. Our construction uses Bestvina-Brady Morse theory, but does not involve branched coverings. 
Yash Lodha
math.GR