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Let $G$ be the group generated by the two transformations $f_1,f_2:\mathbb{R}^2\to \mathbb{R}^2$ given by $f_1(x,y)=(x+1,y)$ and $f_2(x,y)=(x+1,-y)$.

What is the orbit space generated by the action $G\times \mathbb{R}^2\to\mathbb{R}^2$.

I would like to know if there is a way to see the orbit space geometrically.

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    As a start, I claim that $G$ is also generated by $f_1$ and $f_3 = f_2 \circ f_1^{-1}$. (Can you prove this?) It may be easier to see the orbit space using these generators.2012-12-10

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The orbit space is just $[0,1]\times \mathbb{R}^{*}$. I suggest you to think this a while yourself and you can realize it without much difficulty.