Cohomology of group theoretic Dehn fillings II (1908.01290v6)

Abstract: We study the cohomology of group theoretic Dehn fillings. Applying the Cohen-Lyndon property for sufficiently deep Dehn fillings of hyperbolically embedded subgroups $H\hookrightarrow_h G$, obtained by the second named author, we derive a spectral sequence that computes the cohomology of the corresponding Dehn filling quotients $\overline{G}$. As an application, we establish an isomorphism between the relative cohomology of the group pair $(G, H)$ and its sufficiently deep Dehn filling quotient pair $(\overline{G}, \overline{H})$. This allows us to generalise the results of Fujiwara and Manning on simplicial volume of Dehn fillings of hyperbolic manifolds to Dehn fillings of Poincar\'e duality pairs. We also strengthen the results of Olshanskii, Dahmani-Guirardel-Osin and Hull on SQ-universality and common quotients of acylindrically hyperbolic groups by adding cohomological finiteness conditions. We apply these results to obtain hyperbolic and acylindrically hyperbolic quotients with special properties.