On $p$-hyperelliptic involutions of Riemann surfaces

Ewa Tyszkowska

Abstract: A compact Riemann surface $X$ of genus $g>1$ is said to be $p$-hyperelliptic if $X$ admits a conformal involution $\rho$, called a $p$-hyperelliptic involution, for which $X/\rho$ is an orbifold of genus $p$. Here we give a new proof of the well known fact that for $g> 4p+1$, $\rho$ is unique and central in the group of all automorphisms of $X$. Moreover we prove that every two $p$-hyperelliptic involutions commute for $3p+2\leq g\leq 4p+1$ and $X$ admits at most two such involutions if $g> 3p+2$. We also find some bounds for the number of commuting $p$-hyperelliptic involutions and general bound for the number of central $p$-hyperelliptic involutions.

Keywords: $p$-hyperelliptic Riemann surfaces, automorphisms of Riemann surfaces, fixed points of automorphisms

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