Cohomological nonvanishing for modules over discrete groups
Let \(\Gamma\) be a discrete group. A complete resolution over \(\mathbb{Z} \Gamma\) is an acyclic complex \(F_*\) of projective \(\mathbb{Z} \Gamma\)-modules, together with an ordinary projective resolution \(\varepsilon:P_* \to \mathbb{Z}\) such that \(F_*\) and \(P_*\) coincide in sufficiently high dimensions. Let \(F_*\) be a complete resolution over \(\mathbb{Z} \Gamma\) and \(V\) a \(\Gamma\)-module. The Farrell cohomology of \(\Gamma\) with coefficients in \(V\) is defined by the rule \[ \widehat H^* (\Gamma,V) = H^* \bigl(\Hom_{\mathbb{Z} \Gamma} (F_*,V) \bigr). \] Let \({\mathcal X}\) be the class of discrete groups defined as follows: \(\Gamma\in {\mathcal X}\) if there exists a contractible \(\Gamma\)-complex \(X\) with the following properties: (a) \(\Gamma\) includes a normal subgroup \(\Gamma_1\) such that \(\Gamma_1\) acts freely on \(X\) and \(X/ \Gamma_1\) is finite. (b) \(X^H\) is contractible for every finite subgroup \(H\leq \Gamma\). Let \(C^*\) be a cochain \(G\)-complex of finite homological type and \(G\) a finite group. Put \[ \gamma_G (C^*) = |G|\chi \bigl((C^*\otimes \mathbb{Q})^G \bigr)- \chi (C^* \otimes \mathbb{Q}) \] where \(\chi\) is the Euler characteristic. Let \(\Gamma\) be an \({\mathcal X}\)-group, \(X\) an admissible space for \(\Gamma\), \(C^* = C^*(X/ \Gamma_1; \mathbb{Z})\), then \[ \gamma_G(C^*) = |\Gamma: \Gamma_1 |\bigl(\widetilde \chi (\Gamma) - \chi (\Gamma) \bigr), \] the difference between the topological Euler characteristic and the group-theoretical version. Consider \(C^* = \begin{cases} V \quad & \text{for }* =0 \\ 0 \quad & \text{otherwise} \end{cases}\), where \(V\) is a finitely \(\mathbb{Z}\)-generated and \(\mathbb{Z}\)-torsion-free \(G\)-module. Then \[ \gamma_G (C^*) = |G |\text{rank}_\mathbb{Z} V^G-\text{rank}_\mathbb{Z} V. \] Denote this by \(\gamma_G (V)\). In relation the author introduces a numerical invariant \(\gamma_\Gamma(V)\) which determines cohomological nonvanishing under certain hypotheses. The main results of this paper are (a) Theorem 5.23. Let \(\Gamma\) be an \({\mathcal X}\)-group with \(\Gamma_1\) a normal torsion-free subgroup of finite index, \(G= \Gamma/ \Gamma_1\), \(V\) a finitely \(\mathbb{Z}\)-generated \(\mathbb{Z}\)-torsion-free \(\Gamma\)-module. Assume that \(\Gamma\) is not torsion-free and that \(|G|= p^n\) for some prime \(p\). Then either \[ \dim_{\mathbb{F}_p} \widehat H^i (\Gamma,V) \otimes \mathbb{F}_p+ (-1)^{i+1} {\gamma_\Gamma (V) \over |\Gamma: \Gamma_1|} > 0 \quad \text{for every } i\in\mathbb{Z} \] or \[ \widehat H^* (\Gamma,V) = 0 \quad \text{for every} \quad ^* \in\mathbb{Z}. \] (b) Theorem 5.24. Under the hypotheses of the above theorem, precisely one of the following must hold: (1) \(\gamma_\Gamma (V)=0\) and \(V\) is \(\Gamma\)-cohomologically trivial, or (2) \(\gamma_\Gamma (V)=0\) and \(\widehat H^i(\Gamma,V)\neq 0\) for any \(i\in\mathbb{Z}\), or (3) \(\gamma_\Gamma (V)>0\) and \(\widehat H^{2i}(\Gamma,V)\neq 0\) for any \(i\in\mathbb{Z}\), or (4) \(\gamma_\Gamma (V)<0\) and \(\widehat H^{2i-1}(\Gamma,V)\neq 0\) for any \(i\in\mathbb{Z}\). (c) Theorem 4.20. Let \(\Gamma\) be an \({\mathcal X}\)-group, \(\Gamma_1\) a torsion-free normal subgroup of finite index, and \(V\) a finitely generated \(\mathbb{Z}\)-torsion-free \(\Gamma\)-module. Then either \(\widehat H^* (\Gamma,V) \equiv 0\) or \(\widehat H^i (\Gamma,V) \neq 0\) for infinitely many \(i\in \mathbb{Z}\).
scientific article; zbMATH DE number 920031
scientific article