The Segment Lemma

Abstract

This chapter contains the statement and proof of the Segment Lemma. It fixes a parameter A = p/q and set P = 2A/(1 + A) and continues the notation from Chapter 8. The chapter is organized as follows. Section 9.2 explains the existence and image of the anchor point. Section 9.3 explains how to reinterpret the classifying map from Section 8.4 as a map defined on a certain subset of R 3. The technical result, Lemma 9.4, will be useful for the computations in this chapter. Section 9.4 and 9.5 treats the vertical and horizontal cases of the Segment Lemma, respectively.