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how do symmetry groups in the Euclidean space differ from the symmetry groups in the hyperbolic space (in the Poincaré disk) ?

I've been told that in the hyperbolic case one has at disposal a richer "vocabulary" to describe symmetries, but I don't see how, and maybe I misunderstood.

Can anyone give me a hint and/or some source?

Thanks

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# (hyperbolic) symmetry groups

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