Here is a sketch of the following:

**Theorem**. The rational functions doesn’t have Levy cycles.

**Proof.** Suppose it does and denote by the corresponding Levy cycle. Choose the geodesics representatives in . Then the map is the covering map, hence a local isometry and preserves the distances locally. Therefore

where is the component of which is in the same homotpy class with . From the other hand and

and we get that each is strictly shorter than , which is impossible.

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