A characterization of Veech groups in terms of origamis

TL;DR

This paper extends the understanding of Veech groups by characterizing them for flat surfaces with two finite Jenkins-Strebel directions, linking them to origami-like structures and enabling membership decision for matrices.

Contribution

It introduces a combinatorial characterization of Veech groups for certain flat surfaces, broadening the class of surfaces where Veech groups can be explicitly described.

Findings

Veech groups of flat surfaces with two finite Jenkins-Strebel directions can be characterized combinatorially.

Such surfaces decompose into finitely many parallelograms and are of finite analytic type.

The results enable deciding matrix membership in Veech groups for various finite-type flat surfaces.

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 $F_2$. 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.