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We set up a Grothendieck spectral sequence which generalizes the Lyndon–Hochschild–Serre spectral sequence for a group extension  K  ↣  G  ↠  Q  by allowing the normal subgroup  K  to be replaced by a subgroup, or family of subgroups which satisfy a weaker condition than normality. This is applied to establish a decomposition theorem for certain groups as fundamental groups of graphs of Poincaré duality groups. We further illustrate the method by proving a cohomological vanishing theorem which applies for example to Thompson's group  F .

A generalization of the Lyndon–Hochschild–Serre spectral sequence with applications to group cohomology and decompositions of groups