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We compute the higher $\Sigma$-invariants $\Sigma^m(F_{n,\infty})$ of thegeneralized Thompson groups $F_{n,\infty}$, for all $m,n\ge 2$. This extendsthe $n=2$ case done by Bieri, Geoghegan and Kochloukova, and the $m=2$ casedone 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 discussevidence that these bounds are sharp.

On the $\Sigma$-invariants of generalized Thompson groups and Houghton groups