Finiteness properties of subgroups of Houghton groups of full Hirsch length
Finiteness properties of subgroups of Houghton groups of full Hirsch length
In the 1980's K.S. Brown proved that the Houghton group Hn is of type Fn-1 but not FPn. We show that, provided n≥3 , the same conclusion holds for all subgroups G of Hn that are 'large' in the sense that there is an epimorphism G →Zn-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.
math.GR
Martino, Armando
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Kropholler, Peter
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Cox, Charles Garnet
522d9ea0-0890-41c6-848a-bcd0a45e2fca
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Kropholler, Peter
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Cox, Charles Garnet
522d9ea0-0890-41c6-848a-bcd0a45e2fca
Martino, Armando, Kropholler, Peter and Cox, Charles Garnet
(2026)
Finiteness properties of subgroups of Houghton groups of full Hirsch length.
Groups, Geometry and Dynamics.
(In Press)
Abstract
In the 1980's K.S. Brown proved that the Houghton group Hn is of type Fn-1 but not FPn. We show that, provided n≥3 , the same conclusion holds for all subgroups G of Hn that are 'large' in the sense that there is an epimorphism G →Zn-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.
Text
2508.07816v1
- Author's Original
Text
2508.07816v2
- Author's Original
Text
Houghton Full Hirsch Length after GGD suggestions
- Accepted Manuscript
Restricted to Repository staff only until 24 May 2026.
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Text
Houghton Full Hirsch Length revised
- Other
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More information
Accepted/In Press date: 29 January 2026
Keywords:
math.GR
Identifiers
Local EPrints ID: 507207
URI: http://eprints.soton.ac.uk/id/eprint/507207
ISSN: 1661-7207
PURE UUID: e90192ee-3b7a-439e-9550-017d57777e89
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Date deposited: 01 Dec 2025 17:44
Last modified: 29 Apr 2026 01:47
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Author:
Charles Garnet Cox
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