On a property of fundamental groups of graphs of finite groups (Q1181497)

Let \(G\) be a group which acts simplicially on a contractible \(n\)- dimensional simplicial complex \(X\) in such a way that the stabilizer of each simplex is finite. Then there exists a \(\mathbb{Z}\)-free \(\mathbb{Z} G\)-module \(I\) with \(\hbox{proj.dim}_{\mathbb{Z} G}\) \(I\leq n\) and \(H^ 0(G,I)\neq 0\). In the course of the proof the authors also obtain the following result of independent interest: For any group \(G\) the \(G\)-module of bounded functions from \(G\) to \(\mathbb{Z}\) is \(\mathbb{Z} F\)-free for all finite subgroups \(F\) of \(G\).