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 Σ^m(F) and Σ^m(F;ℤ) , the homotopical and homological Bieri–Neumann–Strebel–Renz invariants of F , and show that Σ^m(F) = Σ^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 ).