- 743

- 1

**1. Homework Statement**

Let [itex]\mathbb{H}^2 = \{z=x+iy\in \mathbb{C} | y>0 \}. [/itex] For [itex]a,b,c,d\in \mathbb{R}[/itex] satisfying ad-bc=1 define [itex] T: \mathbb{H}^2 \rightarrow \mathbb{H}^2 [/itex] by

[tex] T(z) = \frac{az+b}{cz+d} [/tex]

Show that T maps that positive y-axis (imaginary axis) to the vertical line [itex] x=\frac{b}{d}, x= \frac{a}{c}[/itex] or a semicircle centred on the x-axis containing both [itex] (\frac{b}{d},0) \text{ and } (\frac{a}{c},0)[/itex]

**3. The Attempt at a Solution**

This seems like it should be fairly easy, but the answer has been eluding me. I began by proceeding as we would in finding the isotropy group, by taking the linear fractional map as a change of variables. Doing this we can set [itex] iy = \frac{aiy+b}{ciy+d} [/itex] and conclude that in general [itex] a=d \text{ and } b=-cy^2 [/itex]. Then using ad-bc=1 we get that [itex] a^2+c^2y^2=1[/itex].

I'm wondering if we need to use anything special about the fact that we can express this mapping as

[tex] \begin{pmatrix} a&b\\ c&d \end{pmatrix} \in SL_2(\mathbb{R})[/tex]

I can't quite seem to figure out where to go from there...