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

TL;DR

This paper computes the higher $oldsymbol{ extSigma}$-invariants of generalized Thompson groups $oldsymbol{F_{n, ext{infty}}}$ for all $oldsymbol{m,n extgreater 1}$ using actions on CAT(0) cube complexes and Morse theory, extending previous results.

Contribution

It introduces a new approach using CAT(0) cube complex actions and Morse theory to compute $oldsymbol{ extSigma}$-invariants for generalized Thompson groups, extending prior work.

Findings

Computed $oldsymbol{ extSigma^m(F_{n, ext{infty}})}$ for all $m,n extgreater 1$

Established lower bounds on $oldsymbol{ extSigma^m(H_n)}$ for Houghton groups

Provided evidence that bounds on Houghton groups are sharp

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 $\textrm{CAT}(0)$ 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.