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Pairs of symmetries of Riemann surfaces. (English) Zbl 0914.30029
Let $$S_g$$ be a compact Riemann surface of genus $$g$$. A symmetry $$T$$ of $$S_g$$ is an anticonformal involution acting on $$S_g$$. The fixed-point set of a symmetry is a collection of disjoint simple closed curves, called the mirrors of the symmetry. The number of mirrors of a symmetry of a surface of genus $$g$$ can be any integer $$k$$ with $$0< k< g+1$$. However, if a Riemann surface $$S_g$$ admits a symmetry $$T_1$$ with $$k$$ mirrors then exist restrictions on the number of mirrors of a different symmetry $$T_2$$ of $$S_g$$. In this work, the authors show these restrictions remarking that the number of such restrictions is small and only occur if one of the symmetries has $$g+1$$ or 0 mirrors.
The $$k$$ mirrors of a symmetry $$T$$ may or may not separate the surface $$S_g$$, into two non-empty components. If the mirrors do separate, then we say that $$T$$ has species $$+k$$, and if the mirrors do not separate then we say that the species is $$-k$$. In the last part of the paper, they obtain a finer classification by investigating which pairs of species can occur for two symmetries $$T_1$$, $$T_2$$ of $$S_g$$. There are many more restrictions than when we just ask for the number of mirrors.

MSC:
 30F10 Compact Riemann surfaces and uniformization 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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