Ellipses in translation surfaces

TL;DR

This paper explores the structure of translation surfaces by characterizing subgroups of the mapping class group that stabilize Teichmueller disks, using immersed ellipses and strips to understand their geometric and topological properties.

Contribution

It introduces a novel characterization of these subgroups through immersed ellipses and strips, and constructs a 2-dimensional cell complex that encodes the surface's geometry and symmetries.

Findings

The space of immersed ellipses/strips forms a non-manifold 2D cell complex.

The topology of this complex determines the translation surface.

The complex's geometry encodes the affine diffeomorphism group.

Abstract

We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed ellipses/strips that meet at least three cone points is naturally a (non-manifold) 2-dimensional cell complex. The topology of this complex and the geometry of its 0-cells determine the translation surface and its affine diffeomorphism group (up to the kernel of the differential).