A spectral sequence for Dehn fillings

TL;DR

This paper introduces a spectral sequence that relates the cohomology of relatively hyperbolic groups before and after Dehn fillings, demonstrating that cohomological dimension remains bounded during the process.

Contribution

It constructs a spectral sequence connecting the cohomology of a group pair and its Dehn filling, showing cohomological dimension does not increase.

Findings

Spectral sequence relates cohomology before and after Dehn fillings.

Cohomological dimension does not increase under sufficiently long Dehn fillings.

Provides new tools for understanding the cohomological behavior of hyperbolic groups.

Abstract

We study how the cohomology of a type $F_\infty$ 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 $F_\infty$, 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.