Paper 3, Section II, D

Groups | Part IA, 2009

Define the cross-ratio [a0,a1,a2,z]\left[a_{0}, a_{1}, a_{2}, z\right] of four points a0,a1,a2,za_{0}, a_{1}, a_{2}, z in C{}\mathbb{C} \cup\{\infty\}, with a0,a1,a2a_{0}, a_{1}, a_{2} distinct.

Let a0,a1,a2a_{0}, a_{1}, a_{2} be three distinct points. Show that, for every value wC{}w \in \mathbb{C} \cup\{\infty\}, there is a unique point zC{}z \in \mathbb{C} \cup\{\infty\} with [a0,a1,a2,z]=w\left[a_{0}, a_{1}, a_{2}, z\right]=w. Let SS be the set of points zz for which the cross-ratio [a0,a1,a2,z]\left[a_{0}, a_{1}, a_{2}, z\right] is in R{}\mathbb{R} \cup\{\infty\}. Show that SS is either a circle or else a straight line together with \infty.

A map J:C{}C{}J: \mathbb{C} \cup\{\infty\} \rightarrow \mathbb{C} \cup\{\infty\} satisfies

[a0,a1,a2,J(z)]=[a0,a1,a2,z]\left[a_{0}, a_{1}, a_{2}, J(z)\right]=\overline{\left[a_{0}, a_{1}, a_{2}, z\right]}

for each value of zz. Show that this gives a well-defined map JJ with J2J^{2} equal to the identity.

When the three points a0,a1,a2a_{0}, a_{1}, a_{2} all lie on the real line, show that JJ must be the conjugation map J:zzˉJ: z \mapsto \bar{z}. Deduce from this that, for any three distinct points a0,a1,a2a_{0}, a_{1}, a_{2}, the map JJ depends only on the circle (or straight line) through a0,a1,a2a_{0}, a_{1}, a_{2} and not on their particular values.

Typos? Please submit corrections to this page on GitHub.