  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.
