Sigma theory for Bredon modules

TL;DR

This paper introduces new invariants for Bredon modules that extend classical invariants, and demonstrates their equivalence in certain classes of groups such as virtually soluble groups and Thompson group F extensions.

Contribution

It develops novel invariants for Bredon modules and establishes their equivalence to classical invariants in specific group categories.

Findings

New invariants for Bredon modules are introduced.

Invariants coincide with classical ones for virtually soluble groups of type FP_infinity.

Invariants also match classical ones for finite extensions of Thompson group F.

Abstract

We develop new invariants similar to the Bieri-Strebel-Neumann-Renz invariants but in the category of Bredon modules (with respect to the class of the finite subgroups of G). We prove that for virtually soluble groups of type FP_{\infty} and finite extension of the Thompson group F the new invariants coincide with the classical ones.