Cohomology of group theoretic Dehn fillings

Abstract

This thesis aims to study cohomology of group theoretic Dehn fillings. For sufficiently deep Dehn fillings of hyperbically embedded subgroups, we first prove that Dehn filling kernels enjoy a particular free product structure, which is termed the Cohen-Lyndon property. We then combine the Cohen-Lyndon property with the Lyndon-Hochschild-Serre spectral sequence to compute cohomology of Dehn filling quotients. As applications, we estimate cohomological dimensions of Dehn fillings quotients, give criterions for Dehn fillings quotients to be of type $FP_{infty}$, and construct useful quotients of acylindrically hyperbolic groups with certain homological properties.