From the second BNSR invariant to Dehn functions of coabelian subgroups

Given a finitely presented group $G$ and a surjective homomorphism $G\to {\mathbb{Z}}^n$ with finitely presented kernel $K$, we give an upper bound on the Dehn function of $K$ in terms of an area-radius pair for $G$. As a consequence we obtain that finitely presented coabelian subgroups of hyperbolic groups have polynomially bounded Dehn function. This generalises results of Gersten and Short and our proof can be viewed as a quantified version of results from Renz’ thesis on the second BNSR invariant.