General origamis and Veech groups of flat surfaces

TL;DR

This paper generalizes the concept of origamis to broader flat surfaces within Teichmüller theory, providing a new parametrization and analyzing Veech groups through combinatorial and linear algebraic methods.

Contribution

It introduces a framework to describe flat surfaces with specified cylinder decompositions and explores the inclusion relations of their Veech groups via coverings.

Findings

Parametrization of flat surfaces with two Jenkins-Strebel directions.

Linear equations realize these surfaces with rectangles of specified moduli.

Inclusion relations of Veech groups under certain coverings.

Abstract

In this century, a square-tiled translation surface (an origami) is intensively studied as an object with special properties of its translation structure and its $SL(2,\mathbb{R})$-orbit embedded in the moduli space. We generalize this concept in the language of flat surfaces appearing naturally in the Teichm\"uller theory. We study the combinatorial structure of origamis and show that a certain system of linear equations realizes the flat surface in which rectangles of specified moduli replace squares of an origami. This construction gives a parametrization of the family of flat surfaces with two finite Jenkins-Strebel directions for each combinatorial structure of two-directional cylinder decomposition. Moreover, we obtain the inclusion of Veech groups of such flat surfaces under a covering relation with specific branching behavior.