The Sigma invariants of Thompson’s group $F$

Abstract

Thompson’s group $F$ is the group of all increasing dyadic $\mathrm{PL}$ 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 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 ${\mathrm{FP}}_m$ (thus certainly not of type $F_m$).