BNSR-Invariants of Surface Houghton Groups

The surface Houghton groups ${\mathcal{H}}_n$ 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 ${\mathrm{\Sigma }}^m(P{\mathcal{H}}_n)$ 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 $P{\mathcal{H}}_n$ 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.