The order of periodic elements of Teichmüller modular groups
Let \(R\) be a Riemann surface with possibly an infinitely generated fundamental group. It is well-known that if \(f\) is a conformal automorphism of \(R\), then \(f\) has finite order and the corresponding element in the mapping class group of \(f\) fixes a point in the Teichmüller space of \(R\). In a previous paper, the author obtained an upper bound for the order of \(f\) in terms of the injectivity radius of \(R\). In the paper under review, the author obtains an extension to the case where \(f\) is quasi-conformal. He shows that in the case where \(f\) fixes the homotopy class of a nontrivial simple closed curve, if the maximal dilatation of \(f\) is smaller than some constant depending on the injectivity radius of \(R\), then the corresponding mapping class is periodic, and he obtains an estimate on the order of the period.
scientific article; zbMATH DE number 2185490