Multiple complexes and gaps in Farrell cohomology (Q1886247)

Let \(\Gamma\) be a group having a torsion free subgroup \(\Gamma'\) of finite index. The cohomological dimension of \(\Gamma'\) does not depend on the choice of \(\Gamma'\) and is called the virtual cohomological dimension of \( \Gamma\). A complete resolution of the trivial \(\Gamma\)-module \(\mathbb Z\) is an acyclic complex \(F\) of projective modules, which coincides with a projective resolution of \(\mathbb Z\) from some degree onwards. Farrell cohomology is the homology of the complex \(\Hom_\Gamma(F,N)\) for a module \(N\). For a group \(G\) let \(r(G)\) be the maximum of the \(p\)-ranks of finite elementary abelian \(p\)-subgroups of \(G\). The main results of the paper are the following. If the cohomological dimension of \(\Gamma'\) is an integer \(n\), and if the trivial \(\Gamma'\)-module \(\mathbb Z\) has a periodic resolution of period \(q\) after \(k\) steps, then the trivial \(\Gamma\)-module \(\mathbb Z\) has a periodic resolution of period \(q\) after \(n+k\) steps. If the virtual cohomological dimension of \(\Gamma\) is an integer \(n\), then the trivial \(\Gamma\)-module \(\mathbb Z\) has a projective resolution being the tensor product of \(r(\Gamma/\Gamma')\) complexes which are periodic resolutions of some period after \(n\) steps. The same holds if one replaces \(r(\Gamma/\Gamma')\) by \(r(\Gamma)\). This last statement uses subtle arguments on cohomology varieties. As an application the author shows, using a spectral sequence argument, that for groups with finite virtual cohomological dimension there is an integer \(r\) so that the vanishing of \(r+1\) consecutive degrees of Farrell cohomology implies vanishing of Farrell cohomology for all degrees.