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We give an explicit construction of a maximal torsion-free finite-indexsubgroup of a certain type of Coxeter group. The subgroup is constructed as thefundamental group of a finite and non-positively curved polygonal complex.First we consider the special case where the universal cover of this polygonalcomplex is a hyperbolic building, and we construct finite-index embeddings ofthe fundamental group into certain cocompact lattices of the building. We showthat in this special case the fundamental group is an amalgam of surface groupsover free groups. We then consider the general case, and construct afinite-index embedding of the fundamental group into the Coxeter group whoseDavis complex is the universal cover of the polygonal complex. All of thegroups which we embed have minimal index among torsion-free subgroups, andtherefore are maximal among torsion-free subgroups.

Maximal torsion-free subgroups of certain lattices of hyperbolic buildings and Davis complexes