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

Abstract

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 $\text{CAT}(0)$ 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 $\text{CAT}(0)$ cube complexes, and discuss evidence that these bounds are sharp.

30 pages, 6 figures

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