A characterization of Veech groups in terms of origamis
Abstract
Schmith\"usen proved in 2004 that the Veech group of an origami is closely related to a subgroup of the automorphism group of the free group F2. This result is significant in the sense that the framework of approachable Veech groups is greatly extended. In this paper, we continue the analysis and consider what kind of settings of flat surfaces allow Veech groups to be characterized combinatorially like origamis. We show that elements in the Veech group of a flat surface with two finite Jenkins-Strebel directions are characterized to allow a concurrence between two `origamis' defined by geodesics in the surface. In the proof we use an observation presented by Earle and Gardiner that a flat surface with two finite Jenkins-Strebel directions is decomposed into a finite number of parallelograms and is proved to be of finite analytic type. Using our results we can decide whether a matrix belongs to the Veech group for various kinds of flat surfaces of finite analytic type.