Finitely presented kernels of homomorphisms from hyperbolic groups onto free abelian groups

For every $m\ge 2$ we produce an example of a non-hyperbolic finitely presented subgroup $H<G$ of a hyperbolic group $G$, which is the kernel of a surjective homomorphism $\phi :G\to {\mathbb{Z}}^m$. The examples we produce are of finiteness type $F_2$ and not $F_3$.