Labeled representations and associated complexes

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.