Non-central fixed point free symmetries of bisymmetric Riemann surfaces

Abstract

We study pairs of symmetries of a Riemann surface of genus $g \geq 2$, whose product has order $n > 2$, assuming that one of them is fixed point free. We start our considerations by giving some bounds for the number of ovals of a symmetry with fixed points and showing their attainment, later we take into account the number of points fixed by the product of the symmetries and we study some of its properties. Finally we deal the problem of finding the maximal possible power of $2$ which can be realized as the order of their product.