The Vertical Lemma

This chapter contains the statement and proof of the Vertical Lemma. Section 10.2 uses symmetry to cut down the amount of work that needs to be done. Section 10.3 translates the picture to prove something more symmetric. The Vertical Lemma associates a point at the center of the unit integer square with a point on the east edge of the square. A more symmetric picture is obtained when working with the centers of the east edges rather than the centers of the squares. Section 10.4 presents some alternative formulas for the map Φ‎: Π‎ → X' and Φ‎: Π‎ → X, which will help with the calculations. Section 10.5 proves the version of the Vertical Lemma for Λ‎'. Section 10.6 makes a careful study of the congruences and shows that the Λ‎'-based result implies the Λ‎-based result.