The Sigma invariants of Thompson’s group $F$

Abstract

Thompson’s group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute Σm(F) and Σm(F;ℤ), the homotopical and homological Bieri–Neumann–Strebel–Renz invariants of F, and show that Σm(F) = Σm(F;ℤ). As an application, we show that, for every m, F has subgroups of type Fm − 1 which are not of type FPm (thus certainly not of type Fm).