Volume 21, issue 5 (2021)

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A spectral sequence for Dehn fillings

Oliver H Wang

Algebraic & Geometric Topology 21 (2021) 2257–2272
Abstract

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

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Keywords
relatively hyperbolic group, Bowditch boundary, group cohomology, Dehn filling, spectral sequence
Primary: 20J06
Secondary: 20F65