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Finiteness properties of Subgroups of Houghton Groups of full Hirsch length
[Submitted on 11 Aug 2025 (v1), last revised 18 Jun 2026 (this v · 2026-06-19 · via math updates on arXiv.org

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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.

Submission history

From: Charles Cox [view email]
[v1] Mon, 11 Aug 2025 09:56:56 UTC (52 KB)
[v2] Mon, 26 Jan 2026 14:14:07 UTC (46 KB)
[v3] Thu, 18 Jun 2026 11:18:38 UTC (46 KB)