Finiteness properties of Subgroups of Houghton Groups of full Hirsch length

TL;DR

The paper extends Brown’s finiteness results for Houghton groups to large subgroups $G\le H_n$ with full Hirsch length, showing that such subgroups are of type $F_{n-1}$ and satisfy max-n, while typically not of type $FP_n$ for $n\ge3$. It develops a structure theorem: $G$ is abstractly commensurable to a restricted multi-wreath product $\mathcal{W}\mathrm{wr}\Gamma$ where the head $\Gamma$ acts on the ray system in a strongly orbit primitive, infinite-orbit fashion, and analyzes the action via a generalized Jordan–Wielandt framework. A multi-wreath embedding (Kaloujnine–Krasner style) and subdirect-product machinery are used alongside BNS invariants and KM1 results to transfer finiteness properties from the head to the whole group. A key contribution is proving finite generation and max-n for full-Hirsch-length subgroups, and delineating when FP properties fail, thereby clarifying the finiteness landscape for large subgroups of $H_n$ within the elementary amenable setting. The results deepen our understanding of cohomological finiteness for these groups and illuminate how large subgroups mirror ambient finiteness characteristics through structured wreath-product decompositions.

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.

Figures (1)

• Figure 1: The ray set $\mathcal{R}_3$ ordered as three rays above and lexicographically below.

Theorems & Definitions (134)

• Theorem 1.1
• Theorem 1.1
• Lemma 1.1
• Theorem 1.1
• Remark
• Theorem 1.1
• Lemma 2.2
• proof
• Definition 2.3
• Remark