Brown's criterion in Bredon homology. (Q384300)

Given a nonnegative integer \(n\), a group \(G\) is said to be of type \(\text{FP}_n\) if the \(\mathbb ZG\)-module \(\mathbb Z\) is of type \(\text{FP}_n\), that is, \(\mathbb Z\) admits a projective \(\mathbb ZG\)-resolution \(P_\bullet\to M\) such that \(P_k\) is finitely generated for \(k\leq n\). \textit{K. S. Brown} [J. Pure Appl. Algebra 44, 45-75 (1987; Zbl 0613.20033)] gave a criterion for testing whether \(G\) is of type \(\text{FP}_n\) in terms of certain reduced homology groups associated to a filtered \(G\)-CW-complex. As suggested by the title, the goal of this paper is to obtain an analogue of Brown's criterion in the context of Bredon modules and homology.NEWLINENEWLINE Given a group \(G\) and a non-empty family \(\mathfrak F\) of subgroups closed under conjugation, one has an associated orbit category \(\mathcal O_{\mathfrak F}G\). A Bredon module over \(\mathcal O_{\mathfrak F}G\) is a functor from the category \(\mathcal O_{\mathfrak F}G\) to the category of Abelian groups. The trivial module \(\underline{\mathbb Z}\) is the functor that takes every object to \(\mathbb Z\) (and every morphism to the identity). A group \(G\) is then said to be of type \(\mathfrak F\)-\(\text{FP}_n\) if \(\underline{\mathbb Z}\) is of type \(\text{FP}_n\) as an \(\mathcal O_{\mathcal F}G\)-module.NEWLINENEWLINE The authors begin with a discussion of basic concepts for Bredon modules and define Bredon homology. The main result is proved in two cases: \(n=0\) and \(n>0\). The \(n=0\) case makes use of a criterion for \(\mathfrak F\)-\(\text{FP}_0\) due to \textit{D. H. Kochloukova} et al. [Bull. Lond. Math. Soc. 43, No. 1, 124-136 (2011; Zbl 1273.20048)]. The \(n>0\) case makes use of work of \textit{C. Martínez-Pérez} and \textit{B. E. A. Nucinkis} [Groups Geom. Dyn. 7, No. 4, 931-959 (2013; Zbl 1295.20055)] who gave a condition for a module to be of type \(\mathfrak F\)-\(\text{FP}_n\) which is an analogue in Bredon homology of the Bieri-Eckmann criterion (in ordinary homology) for being of type \(\text{FP}_n\).NEWLINENEWLINE This criterion in Bredon homology has been used by the second author [Algebr. Geom. Topol. 13, No. 6, 3447-3467 (2013; Zbl 1297.20054)] to show that certain groups have interesting properties with respect to being of type \(\mathfrak F\)-\(\text{FP}_n\).