Planar Families of Discrete Groups

Determining the space of free discrete two generator groups of Möbius 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, nsdc group (see section 2.2 for the definition). Such a group determines three disjoint or tangent planes. We prove that there is a whole family of discrete groups, which we term the nsdc-planar family, that share these planes. From specific information about the nsdc planes which depends on the marking data, we show how to construct all the discrete groups in the nsdc planar family. In particular, we find a set of six real numbers that serve as parameters for this family. We then construct an embedding of our parameters into a classical representation of the full space of free discrete groups as a subset of Ĉ. We see that each planar family fills out a ball of full dimension in the classical embedding. We remark that our construction does not use the usual theory of quasi-conformal deformations of a given group nor does it depend simply on the use of coverings and quotients that give commensurable groups. The paper is organized as follows. In section 2 we give the basic notation and definitions and in section 3 we define the nsdc planar family. Sections 4 and 5 develop computational techniques and in section 6 we construct our new parameters. In section 7 we show how to our parameters are related to the classical parameters. Finally, in section 8 we give a necessary and sufficient condition for a marked group to belong to a given planar family.