Let $\operatorname { SL } ( 2 , \mathbb { R } )$ be the group of $2 \times 2$ matrices with real entries and determinant 1, endowed with the subspace topology of $\mathbb { R } ^ { 4 }$. Consider the continuous map $f : \operatorname { SL } ( 2 , \mathbb { R } ) \longrightarrow \mathbb { C }$ given by $$f \left( \left[ \begin{array} { l l }
a & b \\
c & d
\end{array} \right] \right) = \frac { a i + b } { c i + d }$$ (A) (4 marks) Show that $f$ maps SL$( 2 , \mathbb { R } )$ onto the upper half plane $H = \{ z : \operatorname { Im } ( z ) > 0 \}$. (B) (6 marks) Assume the following two facts: (i) For all $M , N \in \mathrm { SL } ( 2 , \mathbb { R } ) , f ( M ) = f ( N )$ if and only if $M ^ { - 1 } N$ is an orthogonal matrix. (ii) The map $f$ is an open map. Now show that for every compact $K \subseteq H , f ^ { - 1 } ( K )$ is compact.
Let $\operatorname { SL } ( 2 , \mathbb { R } )$ be the group of $2 \times 2$ matrices with real entries and determinant 1, endowed with the subspace topology of $\mathbb { R } ^ { 4 }$. Consider the continuous map $f : \operatorname { SL } ( 2 , \mathbb { R } ) \longrightarrow \mathbb { C }$ given by
$$f \left( \left[ \begin{array} { l l }
a & b \\
c & d
\end{array} \right] \right) = \frac { a i + b } { c i + d }$$
(A) (4 marks) Show that $f$ maps SL$( 2 , \mathbb { R } )$ onto the upper half plane $H = \{ z : \operatorname { Im } ( z ) > 0 \}$.\\
(B) (6 marks) Assume the following two facts:\\
(i) For all $M , N \in \mathrm { SL } ( 2 , \mathbb { R } ) , f ( M ) = f ( N )$ if and only if $M ^ { - 1 } N$ is an orthogonal matrix.\\
(ii) The map $f$ is an open map.
Now show that for every compact $K \subseteq H , f ^ { - 1 } ( K )$ is compact.