Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 23F / b / Solution

Let $q\in {\mathbb{C}}_{\infty }$ and let $G_q$ be its stabilizer subgroup. Choose a sufficiently small disc $U$ about $q$ such thatPart (a) says that $G_q$ is cyclic and supplies a coordinate in which $U\to G_q\backslash U$ is $z\mapsto z^{|G_q|}$. Since the global orbit map identifies no additional points within $U$, this is also a chart on a neighbourhood of the orbit $Gq$ in

Use these quotient charts at every orbit. On overlaps, a local inverse branch of one power map followed by a group translate and the other power map gives the transition function; it is holomorphic. The charts therefore define a conformal structure on $R$, and the local expressions $z\mapsto z^{|G_q|}$ show directly that the quotient map ${\mathbb{C}}_{\infty }\to R$ is holomorphic. This is the finite conformal quotient of a Riemann surface construction.

Solved by gpt-5.6-sol high.