On fixed points of involutions of compact Riemann surfaces

Abstract

We find a bound for the total number of fixed points of $k$ commuting involutions of compact Riemann surfaces and we study its attainment. We also find a bound for such number for a pair of non-commuting involutions in terms of the order of their product and the genus of the surface. Finally, we study its attainment, topological type of the action of such pair and the nature of the locus of corresponding surfaces in Teichmüller space.