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Does there exist a continuous surjective map from $G = GL_2(\Bbb R)$ to the closed unit disc $\{z \in \Bbb C: |z| \le 1\}$ in $\Bbb C$?

I know that $C^{*}$ sits inside $G$ as a subgroup. But it didn't help me. So kindly share some thoughts. Thank you.

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2 Answers 2

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Yes. Let $A=\pmatrix{a&b\\ c&d}$ and denote its Frobenius norm $\sqrt{a^2+b^2+c^2+d^2}$ by $\|A\|_F$. Consider $$ f(A)=\frac{\sqrt{a^2+d^2}}{\|A\|_F}\left(\cos\|A\|_F+i\sin\|A\|_F\right). $$ Clearly $f$ is a continuous function that maps $GL_2(\mathbb R)$ into the closed unit disc on the Argand plane. It is also surjective: for any $0\le r\le1$ and $\theta>0$, we have $f(A)=r(\cos\theta+i\sin\theta)$ when $$ A=\frac{1}{\sqrt{2}}\pmatrix{r\theta&-\sqrt{1-r^2}\theta\\ \sqrt{1-r^2}\theta&r\theta}. $$

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A similar idea as user1551, but a bit simpler example can be constructed. Let $f:GL_2(\mathbb{R})\rightarrow \mathbb{C}$ be defined by $$ f(A)=\frac{b^2}{a^2+b^2} e^{id} $$ where $$ A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}.$$

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