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

We set up a Grothendieck spectral sequence which generalizes the Lyndon--Hochschild--Serre spectral sequence for a group extension $K\mono G\epi Q$ 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 Poincar\'e duality groups. We further illustrate the method by proving a cohomological vanishing theorem which applies for example to Thompson's group $F$.