This dissertation demonstrates the existence of a pair of algebraic and geometric structures on triangles of groups and on quadrilaterals of groups. These structures are an automatic and biautomatic structure. In addition, this paper also discusses the growth function for the quadrilaterals. We show that these groups have these desired structures and discuss what they are. We also give an extraordinary example of a pair of quadrilaterals of groups that are defined nearly identically but do not behave alike. / Ph. D.