Finiteness properties of Subgroups of Houghton Groups of full Hirsch length

TL;DR

The paper extends known finiteness properties of Houghton groups to large subgroups, showing they share similar algebraic finiteness limitations, and explores related generalized wreath products.

Contribution

It demonstrates that large subgroups of Houghton groups have the same finiteness properties as the groups themselves, and introduces a framework for generalized wreath products in this context.

Findings

Large subgroups of Houghton groups are of type F_{n-1} but not FP_n for n ≥ 3.

Generalized permutational wreath products are relevant to the structure of large subgroups.

A generalized Jordan–Wielandt theorem is developed for these wreath products.

Abstract

In the 1980's K.S. Brown proved that the Houghton group $H_n$ is of type $\operatorname{F}_{n-1}$ but not $\operatorname{FP}_n$. We show that, provided $n\ge3$, the same conclusion holds for all subgroups $G$ of $H_n$ that are 'large' in the sense that there is an epimorphism $G\twoheadrightarrow\mathbb{Z}^{n-1}$. Our research leads naturally to the study of generalised permutational wreath products in which the base of the wreath product is a direct product of finite groups which are allowed to vary in isomorphism type from one orbit to another. Such generalised wreath products arise naturally amongst the large subgroups of Houghton groups and are accommodated by a generalised Jordan--Wielandt theorem.