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In [40, Theorems 3.1, 3.2 and 3.3], Jorgensen describes the combinatorial structure of the Ford domain of a quasifuchsian punctured torus group. It was very difficult for the authors to get a conceptual understanding of the statement, because it consists of nine assertions, each of which describes some property of the Ford domain, and it does not explicitly present a topological or combinatorial model of the Ford domain. In this chapter, we construct an explicit model of the Ford domain, and reformulate Jorgensen's theorem in terms of the model. In short, we present a 3-dimensional picture to Jorgensen's theorem. We note that this chapter is essentially equal to the announcement [10].

The topological once-punctured torus T , the 4-times punctured sphere S and the (2, 2, 2, ∞)-orbifold O are commensurable, and are called Fricke surfaces (see [74]). In this chapter, we give a detailed study of the fundamental groups of Fricke surfaces and their representations to PSL (2, ℂ).

In this chapter, we introduce the notation which we use to reformulate the main theorems in Chap. 6. As explained in Sect. 2.4, the infinite broken line in ℂ obtained by successively joining the centers c ( ρ ( P _ j )) of the isometric circles I ( ρ ( P _ j )), where { P _ j } is a sequence of elliptic generators, recovers the type-preserving representation ρ . Moreover, this broken line plays a key role in the description of the combinatorial structure of the Ford domain in the case ρ is quasifuchsian. Thus we introduce, in Sect. 3.1, the notation L ( ρ, σ ) to represent the broken line, where σ is the triangle of the Farey triangulation spanned by the slopes of { P _ j }. Then we introduce the concept for a Markoff map to be upward at σ (Definition 3.1.3), and show that precisely one Markoff map among the four Markoff maps inducing a given representation is upward (Lemma 3.1.4). This concept is used in Sect. 4.2 to define the side parameter.

The essential ingredient of Jorgensen's work in [40] is a detailed analysis of how the pattern of isometric hemispheres bounding the Ford domain change as one varies the group. This idea can be found in his preceding work [39] on the infinite cyclic Kleinian groups. (See the work [25] due to Drumm and Poritz for its detailed exposition and generalization.) In this chapter we first describe the “chain rule for isometric circles” (Lemma 4.1.2), which affords a foundation on the analysis, and then we introduce the key notion of Jorgensen's side parameter (Definition 4.2.9) and prove various of its properties.

In this chapter, we give a detailed study of special examples.

In this chapter, we give a “2-dimensional” reformulation of the Main Theorem 1.3.5 and present a route map of its proof.

This chapter is devoted to the proof of Proposition 6.2.1 (Openness). The main ingredient of the proof is the study of the behavior of hidden isometric hemispheres , namely those isometric hemispheres which intersect the Ford domain but do not support 2-dimensional faces of the Ford domain (Definition 7.1.1).

In this chapter, we study what happens at the limit of a sequence { ρ _ n } = {( ρ _ n , ν _ n )} of good labeled representations.

The purpose of this chapter is to prove Proposition 6.2.6 (Unique realization), which implies the bijectivity of the map μ _2 : J [ QF ] → ℍ^2 × ℍ^2. To this end, we first make a careful study of the algebraic curves in the algebraic surface Φ ≅ {( x , y , z ) ∈ ℂ^3 I  x ^2 +  y ^2 +  z ^2 =  xyz } determined by the equations in Definition 4.2.19, and find the irreducible components which contain the geometric roots (Definition 9.1.2) for a given label ν  = ( ν ^−, ν ^+) ∈ ℍ^2 × ℍ^2 (Lemmas 9.1.8 and 9.1.12). We also observe that the number of the algebraic roots for ν is finite (Proposition 9.1.13). Thus the problem is how to single out the geometric roots among the algebraic roots. Our answer is to appeal to the idea of the geometric continuity . By using the idea, we show that all geometric roots for a given label ν are obtained by continuous deformation of the unique geometric root for a special label corresponding to a fuchsian group. This implies the desired result that each label has the unique geometric root. To realize this idea, we introduce the concept of the “geometric degree” d _ G ( ν ) of a label ν , and then show that d _ G ( ν ) = 1 for every ν by using the argument of geometric continuity (Proposition 9.2.3).