Fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements

We determine the biholomorphic fiber preserving isomorphisms of fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements. We show that except in some special cases a biholomorphic fiber preserving isomorphism between two Bers fiber spaces is always an allowable mapping. We find that the situation is different for Teichmüller curves, showing that in general there are some other biholomorphic fiber preserving isomorphisms between Teichmüller curves besides the allowable mappings. Research supported by the National Natural Science Foundation of China.