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Is it possible that a point on the unit circle which is an ordinary point (that is, a point which is not a limit point of any set of the form $\Gamma z$ for $|z| <1$) for a Fuchsian group $\Gamma$ (acting on the Poincare hyperbolic disk) is fixed by a nontrivial element of that group? What if it is also a vertex at infinity for a Dirichlet domain for the group?

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I first thought the question might be a bit tricky and that one would need to go through the classification of Fuchsian groups (elementary, first kind, second kind...) but actually I think it's pretty straightforward to answer that no, this never happens!

Since elliptics have no fixed points on $\partial \mathbb{D}$, you only need to look at fixed points of parabolic or hyperbolic elements of $\Gamma$. But in both cases, it is easy to see that those are limit points, just using powers of your parabolic/hyperbolic element.

Is that convincing?

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  • $\begingroup$ Ah! yes, thank you. $\endgroup$
    – Klam
    Commented Jan 6, 2014 at 18:30

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