$\mathrm{SL}_2(\mathbb{R})$-dynamics on the moduli space of one-holed tori

Abstract

We study the $\mathrm{SL}_2(\mathbb{R})$-action on the moduli space of (triangulable) dilation tori with one boundary component. We prove that every orbit is either closed or dense, and that every orbit of the Teichmuller flow escapes to infinity.