2025/08/11 by Cox, Charles, Kropholler, Peter, Martino, Armando
#18G10 #20J05 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2508.07816
For n∈ \1,2,3, …\, the nth Houghton group Hn is the group of those permutations permutations g of the ray set \1, …, n\ × ℕ that are eventually translations along each ray in the sense that there exists j0 depending on g and a vector (t1,...,tn)∈ℤn also depending on g, such that for all 1≤ i≤ n, and all j≥ j0, (i,j)g=(i,j+ti) For all n≥ 1, Hn affords an epimorphism to ℤn-1 whose kernel is the set of finitary permutations of the ray set. K. S. Brown showed that Hn has type Fn-1 but not type FPn, meaning that Hn 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≥3, the same conclusion holds for subgroups G of Hn that are large in the sense that there is an epimorphism G\twoheadrightarrowℤn-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.