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What is the generator of translation in the Beltrami-Klein model of the hyperbolic plane?

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closed as too localized by Ryan Budney, José Figueroa-O'Farrill, Will Jagy, Richard Kent, Andy Putman Jun 3 '11 at 4:45

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I've voted to close. is a more appropriate forum for your question. – Ryan Budney Jun 2 '11 at 7:05
I've also voted to close. The notion of "the generator of translation" also makes no sense, as r0b0t's answer below suggests. – José Figueroa-O'Farrill Jun 2 '11 at 13:29

Hyperbolic plane is a homogeneous space $G/H$ where $G$ acts by isometries and so any reasonable defined `translation' is given by the natural left action of $G$. Decide what elements of $G$ will you call translations and then for any abstract generator $X\in\mathfrak{g}$ such that $\exp{tX}$ is translation, compose the natural left action with diffeomorphism of $G/H$ to your favorite model. Differentiate the resulting map at zero and you are done.

If I am not mistaken, this question is more suitable for math.SE.

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