Sigma theory for Bredon modules

Abstract

We develop new invariants ${\underline{\Sigma }}^m(G,\underline{A})$ similar to the Bieri–Strebel–Neumann–Renz invariants ${\Sigma }^m(G,A)$ but in the category of Bredon modules $\underline{A}$ (with respect to the class of the finite subgroups of $G$). We prove that for virtually soluble groups of type ${\mathrm{FP}}_{\infty }$ and finite extension of the Thompson group $F$ we have ${\underline{\Sigma }}^{\infty }(G,\underline{\mathbb{Z}})={\Sigma }^{\infty }(G,\mathbb{Z})$.