A spectral sequence for Dehn fillings

Abstract

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