Finiteness properties of Subgroups of Houghton Groups of full Hirsch length

Abstract: For $n\in {1,2,3, \ldots}$, the $n$th Houghton group $H_n$ is the group of those permutations permutations $g$ of the ray set ${1, \ldots, n} \times \mathbb{N}$ that are eventually translations along each ray in the sense that there exists $j_0$ depending on $g$ and a vector $(t_1,...,t_n)\in\mathbb{Z}^{n}$ also depending on $g$, such that for all $1\le i\le n$, and all $j\ge j_0$, $$(i,j)g=(i,j+t_i)$$ For all $n\ge 1$, $H_n$ affords an epimorphism to $\mathbb{Z}^{n-1}$ whose kernel is the set of finitary permutations of the ray set. K. S. Brown showed that $H_n$ has type $\operatorname{F}_{n-1}$ but not type $\operatorname{FP}_n$, meaning that $H_n$ has an Eilenberg--MacLane space with finite $(n-1)$-skeleton but does not have an Eilenberg--MacLane with finite $n$-skeleton. We show that, provided $n\ge3$, the same conclusion holds for subgroups $G$ of $H_n$ that are large in the sense that there is an epimorphism $G\twoheadrightarrow\mathbb{Z}^{n-1}$. Our results depend on a generalised form of the Jordan--Wielandt theorem for intransitive permutation groups, as well as a structural analysis of permutational wreath products that allows for distinct chandelier groups associated with distinct orbits when the permutation representation is intransitive.