Planar families of discrete groups

Determining the space of free discrete two generator groups of M\"obius transformations is an old and difficult problem. In this paper we show how to construct large balls of full dimension in this space. To do this, we begin with a marked discrete group of non-separating disjoint circle type. Such a group determines three disjoint or tangent planes. We prove that there is a whole family of discrete groups that share these planes. We find a set of six real numbers that serve as parameters for this family and construct an embedding of our parameters into a classical representation of the full space of free discrete groups. We see that each planar family fills out a ball of full dimension in the classical embedding.