The Sigma Invariants of Thompson's Group F

TL;DR

This paper computes the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of Thompson's group F, revealing subgroup structures related to finiteness properties.

Contribution

It provides the first detailed computation of Sigma invariants for Thompson's group F, establishing their equality and implications for subgroup finiteness types.

Findings

Sigma^m(F) equals Sigma^m(F;Z) for all m

Thompson's group F has subgroups of type F_{m-1} not of type F_m

Advances understanding of subgroup finiteness properties in F

Abstract

Thompson's group F is the group of all increasing dyadic piecewise linear homeomorphisms of the closed unit interval. We compute Sigma^m(F) and Sigma^m(F;Z), the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F, and we show that Sigma^m(F) = Sigma^m(F;Z). As an application, we show that, for every m, F has subgroups of type F_{m-1} which are not of type F_{m}.