  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 $\Sigma^{m}(P\mathcal{H}_{n})$ 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.
