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Abstract

We study the upper bounds for the total number of ovals of two symmetries of a Riemann surface of genus $g$, whose product has order $n$. We show that the natural bound coming from Bujalance, Costa, Singerman and Natanzon's original results is attained for arbitrary even $n$, and in case of $n$ odd, there is a sharper bound, which is attained. We also prove that two $(M-q)$- and $(M-q')$-symmetries of a Riemann surface $X$ of genus $g$ commute for $g\geq q+q'+1$ (by $(M-q)$-symmetry we understand a symmetry having $g+1-q$ ovals) and we show that actually, with just one exception for any $g>2$, $q+q'+1$ is the minimal lower bound for $g$ which guarantees the commutativity of two such symmetries.