| An origami is a combinatorial object obtained by gluing Euclidean squares along their edges according to a few simple rules. The resulting closed surface X carries a natural structure as a translation surface. Furthermore the tiling by squares defines a covering p: X -> E of the torus E ramified over at most one point.
This setting gives a holomorphic and isometric embedding of the Teichmueller space T_{1,1} of once punctured tori into the Teichmueller space T(X). Its image in T(X) is a special case of a Teichmueller disk; the projection into moduli space is a Teichmueller curve. Teichmueller curves have recently attracted remarkable attention in different fields as dynamical systems and algebraic geometry.
We describe an analogous construction in the Culler-Vogtmann outer space, which can be considered as the Teichmueller space for metric graphs, and transfer results between these two settings. |
