A spectral sequence for Dehn fillings

We study how the cohomology of a type $F_{\mathrm{\infty }}$ relatively hyperbolic group pair $(G,\mathcal{P})$ changes under Dehn fillings (i.e. quotients of group pairs). For sufficiently long Dehn fillings where the quotient pair $(\bar{G},\bar{\mathcal{P}})$ is of type $F_{\mathrm{\infty }}$, we show that there is a spectral sequence relating the cohomology groups $H^i(G,$\mathcal{P}$;$\mathbb{Z}$ G)$and$H^i\left(\bar{G},\bar{$\mathcal{P}$};$\mathbb{Z}$\bar{G}\right)$. As a consequence, we show that essential cohomological dimension does not increase under these Dehn fillings.