Paper 3, Section II, E

Groups | Part IA, 2012

Let GG be SL2(R)S L_{2}(\mathbb{R}), the groups of real 2×22 \times 2 matrices of determinant 1 , acting on C{}\mathbb{C} \cup\{\infty\} by Möbius transformations.

For each of the points 0,i,i0, i,-i, compute its stabilizer and its orbit under the action of GG. Show that GG has exactly 3 orbits in all.

Compute the orbit of ii under the subgroup

H={(ab0d)a,b,dR,ad=1}G.H=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d=1\right\} \subset G .

Deduce that every element gg of GG may be expressed in the form g=hkg=h k where hHh \in H and for some θR\theta \in \mathbb{R},

k=(cosθsinθsinθcosθ)k=\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)

How many ways are there of writing gg in this form?

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