On commutativity and ovals for a pair of symmetries of a Riemann surface
Volume 109 / 2007
Colloquium Mathematicum 109 (2007), 61-69 MSC: Primary 30F50; Secondary 14H37. DOI: 10.4064/cm109-1-5
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.