Let be , the groups of real matrices of determinant 1 , acting on by MÃ¶bius transformations.
For each of the points , compute its stabilizer and its orbit under the action of . Show that has exactly 3 orbits in all.
Compute the orbit of under the subgroup
Deduce that every element of may be expressed in the form where and for some ,
How many ways are there of writing in this form?