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This is the second part of our study of epimorphisms with source a thick generalized -gon and target a thin generalized -gon. We classify the case when the polygons are finite (in the first part [15] we handled the cases and ). Then we show that the infinite case is very different, and construct examples which strongly differ from the finite case. A number of general structure theorems are also obtained, and we also take a look at the infinite case for general gonality.

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