On commutativity and ovals for a pair of symmetries of a Riemann surface

Ewa Kozłowska-Walania¹

¹ Institute of Mathematics Gdańsk University Wita Stwosza 57 80-952 Gdańsk, Poland

Colloquium Mathematicum, Tome 109 (2007) no. 1, pp. 61-69.

Résumé

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.

Détail

DOI : 10.4064/cm109-1-5

Keywords: study upper bounds total number ovals symmetries riemann surface genus whose product has order natural bound coming bujalance costa singerman natanzons original results attained arbitrary even odd there sharper bound which attained prove m q m q symmetries riemann surface genus commute geq m q symmetry understand symmetry having q ovals actually just exception minimal lower bound which guarantees commutativity symmetries