Pairs of symmetries of Riemann surfaces.

*(English)*Zbl 0914.30029Let \(S_g\) 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.

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.

Reviewer: E.Bujalance (Madrid)