The surface Houghton groups Hₙ are a family of groups generalizing Houghton groups Hₙ, which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants Σᵐ(PHₙ) 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 Hₙ and PHₙ 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.
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Torgerson et al. (2024) studied this question.
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