We compute the higher $\Sigma$-invariants $\Sigma^m(F_{ n,\infty})$ of the generalized Thompson groups $F_{ n,\infty}$, for all $m,n\ge 2$. This extends the $n=2$ case done by Bieri, Geoghegan and Kochloukova, and the $m=2$ case done by Kochloukova. Our approach differs from those used in the $n=2$ and $m=2$ cases; we look at the action of $F_{ n,\infty}$ on a $\textrm{ CAT}(0)$ cube complex, and use Morse theory to compute all the $\Sigma^m(F_{ n,\infty})$. We also obtain lower bounds on $\Sigma^m(H_n)$, for the Houghton groups $H_n$, again using actions on $\textrm{ CAT}(0)$ cube complexes, and discuss evidence that these bounds are sharp.
Categories: math.GR math.GT