The order of conformal automorphisms of Riemannian surfaces of infinite type (Q1420217)

Let \(R\) be a hyperbolic Riemann surface, i.e, the quotient \(R = \mathbb{H}/\Gamma\) where \(\mathbb{H}\) is the upper complex halfplane and \(\Gamma\) is a torsion-free Fuchsian group. Let \(M_R\) be a positive real number greater than the injectivity radius at every point of \(R\). The author shows that the order of any conformal automorphism \(f\) of \(R\) with finite order which fixes a compact subset of \(R\), is upperly bounded by a function depending only on \(M_R\). This result extends to the noncompact case a classical result by Wiman: the order of an automorphism \(f\) of a compact genus \(g\) Riemann surface is upperly bounded by \(2(2g+ 1)\).