Groups, periodic planes and hyperbolic buildings

Abstract

Using methods of combinatorial group theory, we give an elementary construction of polyhedra whose links are (not necessarily isomorphic) connected bipartite graphs. In particular, we construct polyhedra whose links are generalized m -gons. Polyhedra of this type are interesting because their universal coverings are two-dimensional hyperbolic buildings with different links. We show that the fundamental groups of some of our polyhedra contain surface groups. The presentation of the results is given in the language of combinatorial group theory.