The order of conformal automorphisms of Riemann surfaces of infinite type

Abstract

Let $R$ be a Riemann surface of infinite type such that the injectivity radius at any point in $R$ is less than a positive constant $M$, and $f$ a conformal automorphism of $R$ fixing a compact subset in $R$. We show that the order of $f$ is less than a certain constant depending on $M$.