On periodic Teichmüller disks of modular transformations
This paper explores what could be called the `affine realization problem': given an element \(\phi\) of the mapping class group \(\Gamma_g\) of a closed surface of genus \(g\) (or more generally a subgroup) when does it stabilize a Teichmüller disk, that is when can it be made to act by affine diffeomorphisms of a singular flat metric arising from a holomorphic differential? Thurston's classification of mapping classes into elliptic, hyperbolic, and reducible elements gives three flavors of this problem: it can be shown that any elliptic element does preserve a flat structure (in fact, \textit{A. Hadari} [``Translation surfaces with finite Veech groups'', \url{arXiv:1005.3516}] shows that any finite subgroup of \(\mathrm{GL}(2, \mathbb R)\) arises as a group of affine diffeormorphisms), and similarly for hyperbolic elements the Teichmüller disk defined by the quadratic differential associated to the hyperbolic element is stabilized. For parabolic elements (and groups generated by them) the situation is more complicated. \textit{A. Marden} and \textit{H. Masur} [Math. Scand. 36, 211--228 (1975; Zbl 0318.32019)] showed that Dehn twists stabilize disks, and Thurston showed that the group generated by pairs of multitwists around curves that fill the surface also stabilizes disks. The main result in this paper is that a parabolic transformation either has no periodic Teich-müller disks (that is, no power stabilizes a disk) or it has infinitely many, in which case the infinitely many Teichmüller disks can be chosen to cover infinitely many arithmetic Teichmüller curves in the Riemann moduli space \(\mathcal M_g\).
scientific article; zbMATH DE number 6361345