BNSR-invariants of surface Houghton groups

Torgerson, Noah; West, Jeremy

doi:10.2140/agt.2025.25.4897

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ISSN (electronic): 1472-2739

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BNSR-invariants of surface Houghton groups

Noah Torgerson and Jeremy West

Algebraic & Geometric Topology 25 (2025) 4897–4919

DOI: 10.2140/agt.2025.25.4897

Abstract

The surface Houghton groups $ℋ_{n}$ are a family of groups generalizing Houghton groups $H_{n}$ , which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants $Σ^{m} (P ℋ_{n})$ of their intersection with the pure mapping class group. To do so, we prove that the associated Stein–Farley cube complex is $CAT (0)$ , and we adapt Zaremsky’s method for computing the BNSR-invariants of the Houghton groups. As a consequence, we give a criterion for when subgroups of $H_{n}$ and $P ℋ_{n}$ having the same finiteness length as their parent group are finite index. We also discuss the failure of some of these groups to be co-Hopfian.

Keywords

surface Houghton groups, asymptotic rigidity, medium mapping class groups, $\mathrm{CAT}(0)$ cube complexes, BNSR-invariants, BNS-invariants, sigma invariants

Mathematical Subject Classification

Primary: 20F65, 57K20, 57M07

References

Publication

Received: 20 March 2024

Revised: 7 June 2024

Accepted: 17 September 2024

Published: 20 November 2025

Authors

Noah Torgerson
Department of Mathematics University of Oklahoma Norman, OK United States

Jeremy West
Department of Mathematics University of Oklahoma Norman, OK United States

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Volume 25, issue 8 (2025)