The BNSR-invariants of the Houghton groups, concluded

Abstract

We give a complete computation of the BNSR-invariants $\Sigma^m(H_n)$ of the Houghton groups $H_n$. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated $CAT(0)$ cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each $1\le m\le n-1$, $H_n$ admits a map onto $\mathbb{Z}$ whose kernel is of type $F_{m-1}$ but not $F_m$, and moreover no such kernel is ever of type $F_{n-1}$.