  We provide a condition on the links of polygonal complexes that is sufficient
to ensure groups acting properly discontinuously and cocompactly on such
complexes contain a virtually free codimension-1 subgroup. We provide stronger
conditions on the links of polygonal complexes, which are sufficient to ensure
groups acting properly discontinuously and cocompactly on such complexes act
properly discontinuously on a CAT(0) cube complex. If the group is hyperbolic
then this action is also cocompact, hence by Agol's Theorem the group is
virtually special (in the sense of Haglund-Wise); in particular it is linear
over Z. We consider some applications of this work. Firstly, we consider the
groups classified by [KV10] and [CKV12], which act simply transitively on
CAT(0) triangular complexes with the minimal generalized quadrangle as their
links, proving that these groups are virtually special. We further apply this
theorem by considering generalized triangle groups, in particular a subset of
those considered by [CCKW20].
