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bnsr-invariant

Proceedings of the Edinburgh Mathematical Society (1)

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1 results

The BNSR-invariants of the Houghton groups, concluded
Part of
- Special aspects of infinite or finite groups
- Low-dimensional topology
Matthew C. B. Zaremsky
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 63 / Issue 1 / February 2020

Published online by Cambridge University Press:

15 July 2019, pp. 1-11
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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.