A new spectral sequence and an algebraic excision theorem control the cohomology of Dehn fillings of groups with hyperbolically embedded subgroups, with applications to Poincaré duality, simplicial volume, and acylindrically hyperbolic quotients.
A spectral sequence for Dehn fillings
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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.
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math.GR 1years
2019 1verdicts
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Cohomology of group theoretic Dehn fillings II
A new spectral sequence and an algebraic excision theorem control the cohomology of Dehn fillings of groups with hyperbolically embedded subgroups, with applications to Poincaré duality, simplicial volume, and acylindrically hyperbolic quotients.