The $\Sigma$-invariants of Thompson's group $F$ via Morse theory

TL;DR

This paper re-derives the BNSR-invariants of Thompson's group F using geometric methods, specifically leveraging the Stein--Farley CAT(0) cube complex, providing an alternative to previous algebraic computations.

Contribution

It introduces a geometric approach to compute the BNSR-invariants of Thompson's group F, complementing prior algebraic methods.

Findings

Recomputed BNSR-invariants using geometric techniques

Utilized Stein--Farley CAT(0) cube complex

Provided an alternative proof to existing results

Abstract

Bieri, Geoghegan and Kochloukova computed the BNSR-invariants $\Sigma^m(F)$ of Thompson's group $F$ for all $m$. We recompute these using entirely geometric techniques, making use of the Stein--Farley CAT(0) cube complex on which $F$ acts.