Subgroup posets, Bredon cohomology and equivariant Euler characteristics. (Q2847039)

Let \(G\) be a discrete group and \(\mathcal F\) be the family of finite subgroups of \(G\). The importance of the poset \(\mathcal F\) has been observed in many instances. The author explores properties of \(\mathcal F\) and constructs suitable Bredon projective resolutions for the trivial module in terms of projective resolutions of the chain complexes associated to some posets of subgroups of \(G\). Let \(\underline EG\) denote a model for the universal space for proper actions for \(G\). The author gives a formula to compute the equivariant Euler class of \(\underline EG\) when \(G\) is virtually solvable of type \(\text{FP}_\infty\).NEWLINENEWLINE The main results are the following: Theorem A. Let \(G\) be a group with a bound on the \(\mathcal H\)-lengths of the subgroups \(H\in\mathcal H\). Then NEWLINE\[NEWLINE\text{cd}_{\mathcal H}G=\max_{H\in\mathcal H}\text{pd}_{WH}\Sigma\widetilde{\mathcal H}_{H\bullet},NEWLINE\]NEWLINE where \(\mathcal H_{H\bullet}\) denotes the augmented chain complex associated to the poset \(\mathcal H\), \(WH=N_G(H)/H\), and \(\text{pd}_{WH}\) denotes the projective dimension of a \(WH\)-complex. In this setup \(\mathcal H\) denotes a class of subgroups of \(G\) closed under conjugation.NEWLINENEWLINE In the case of taking the family of finite subgroups of \(G\) the author proves the following Theorem B. Let \(G\) be a group with a bound on the orders of the finite subgroups and \(\underline{\text{cd}}\,G<\infty\). Assume that there is an order reversing function \(\ell\colon\mathcal F\to\mathbb Z\) such that for each finite subgroup \(H<G\), one has \(\text{pd}_{WK}B(WH)\leq\ell(H)\) and either \(\ell(K)<\ell(H)\) for any \(H<K\), or \(|\mathcal F_H|\simeq *\). Then \(\underline{\text{cd}}\,G\leq\ell(1)\). Here \(\underline{\text{cd}}\,G\) stands for the virtual cohomological dimension of \(G\).