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.
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© 2007 Springer-Verlag Berlin Heidelberg
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(2007). Labeled representations and associated complexes. In: Punctured Torus Groups and 2-Bridge Knot Groups (I). Lecture Notes in Mathematics, vol 1909. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71807-9_3
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DOI: https://doi.org/10.1007/978-3-540-71807-9_3
Publisher Name: Springer, Berlin, Heidelberg
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