  We give an explicit construction of a maximal torsion-free finite-index
subgroup of a certain type of Coxeter group. The subgroup is constructed as the
fundamental group of a finite and non-positively curved polygonal complex.
First we consider the special case where the universal cover of this polygonal
complex is a hyperbolic building, and we construct finite-index embeddings of
the fundamental group into certain cocompact lattices of the building. We show
that in this special case the fundamental group is an amalgam of surface groups
over free groups. We then consider the general case, and construct a
finite-index embedding of the fundamental group into the Coxeter group whose
Davis complex is the universal cover of the polygonal complex. All of the
groups which we embed have minimal index among torsion-free subgroups, and
therefore are maximal among torsion-free subgroups.
