This paper aims at 2-dimensional complexes of nonpositive curvature. Classical examples are the regular tessellations \(\{p, q\}\), \({1\over p}+ {1\over q}\leq {1\over 2}\). In Chapters 1 and 2 the authors construct many \(k\)-complexes (consisting of \(k\)-gons) such that the link of each vertex is the 1-skeleton of an \(n\)-simplex or of an \(n\)-cube. In particular, regular complexes of this type and their groups are studied. Classification results are obtained, for \(k\geq 6\) and \(n\geq 4\) there is a uniqueness result. Chapter 3 of the paper gives an interesting relationship with the word problem and regular formal languages.