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The BNSR-invariants of the Houghton groups, concluded

Part of: Low-dimensional topology Special aspects of infinite or finite groups

Published online by Cambridge University Press:  15 July 2019

Matthew C. B. Zaremsky
Matthew C. B. Zaremsky*
Affiliation:
Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY12222, USA ([email protected])
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Abstract

We give a complete computation of the Bieri–Neumann–Strebel–Renz invariants Σm(Hn) of the Houghton groups Hn. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated CAT (0) cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each 1 ≤ m ≤ n − 1, Hn admits a map onto ℤ whose kernel is of type Fm−1 but not Fm; moreover, no such kernel is ever of type Fn−1.

Keywords

Houghton groupBNSR-invariantfiniteness propertiescube complexCAT(0)

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 57M07: Topological methods in group theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 1 - 11
Copyright
Copyright © Edinburgh Mathematical Society 2019

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