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Families of Riemann surfaces over the punctured disk. (English) Zbl 0734.30039
Holomorphic families of compact Riemann surfaces of genus p over the once puntured unit disk are constructed whose relative canonical line bundles (i.e. restricting to the canonical line bundle of each surface of the family) admit no roots of order n, for each \(n>2\) and \(p\geq 3\) (one can assume there that n divides 2p-2 because otherwise the canonical line bundle of a Riemann surface itself admits no root of order n). The monodromy groups of these families are generated by certain powers of commuting Dehn twists around simple closed curves. It is also shown that only positive powers of commuting negative Dehn twists occur as the monodropy of a family of Riemann surfaces over the punctured disk, and a generalization is announced characterizing those mapping classes of infnite order which occur as the monodromy of such families.

30F99 Riemann surfaces
30F60 Teichm├╝ller theory for Riemann surfaces
30F10 Compact Riemann surfaces and uniformization
Dehn twists
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