Sigma Invariants of Direct Products of Groups

TL;DR

This paper proves the Product Conjecture for Sigma invariants over a field and under certain conditions over Z, leading to new subgroup properties of Thompson's group F.

Contribution

It extends the validity of the Product Conjecture for Sigma invariants over Z under specific hypotheses, despite known counterexamples.

Findings

Proves the Product Conjecture over a field.

Establishes conditions where the conjecture holds over Z.

Applies results to analyze subgroups of Thompson's group F.

Abstract

The Product Conjecture for the homological Bieri-Neumann-Strebel-Renz invariants is proved over a field. Under certain hypotheses the Product Conjecture is shown to also hold over Z, even though D. Schuetz has recently shown that the Conjecture is false in general over Z. Our version over Z is applied in a joint paper with D. Kochloukova to derive new information about subgroups of Thompson's group F, namely that F has subgroups F_m which are not of type F_{m+1}.