Paper 3, Section II, D

Groups | Part IA, 2019

Let M\mathcal{M} be the group of Möbius transformations of C{}\mathbb{C} \cup\{\infty\} and let SL2(C)\mathrm{SL}_{2}(\mathbb{C}) be the group of all 2×22 \times 2 complex matrices of determinant 1 .

Show that the map θ:SL2(C)M\theta: \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \mathcal{M} given by

θ(abcd)(z)=az+bcz+d\theta\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)(z)=\frac{a z+b}{c z+d}

is a surjective homomorphism. Find its kernel.

Show that any TMT \in \mathcal{M} not equal to the identity is conjugate to a Möbius map SS where either Sz=μzS z=\mu z with μ0,1\mu \neq 0,1 or Sz=z+1S z=z+1. [You may use results about matrices in SL2(C)\mathrm{SL}_{2}(\mathbb{C}) as long as they are clearly stated.]

Show that any non-identity Möbius map has one or two fixed points. Also show that if TT is a Möbius map with just one fixed point z0z_{0} then Tnzz0T^{n} z \rightarrow z_{0} as nn \rightarrow \infty for any zC{}z \in \mathbb{C} \cup\{\infty\}. [You may assume that Möbius maps are continuous.]

Typos? Please submit corrections to this page on GitHub.