The Sigma invariants of Thompson's group F

Thompson's group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute † m .F / and † m .F I Z/, the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F , and show that † m .F / D † m .F I Z/ .A s 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).