The Eleventh Week: The Group of Covering Transformations

Let f : D’ → D be a covering. We say that points P₁ and P₂ of D’ are conjugate if f(P₁) = f(P₂). The number of points which are conjugate to P₁ (including P₁ itself) is equal to n = deg(f). Similarly, we say that curves C₁’and C₂’ are conjugate if they are lifts of the same curve. In order to construct a curve starting at P₂ which is conjugate to a curve C₁’ starting at P₁, first we project C₁’ to make a curve C. Then take the lift of C which starts at P₂. The number of curves conjugate to C_l’ (including C_l’ itself) in D’ is also equal to n = deg(f).