
## 5. Infinite translation surfaces

1. #### Problem 5.1.

Which mapping classes are realized by affine automorphisms on some translation surface?
1. Remark. We can ask the same question, but for Penner’s construction: Given a mapping class that is obtained by Penner’s construction, does it fix some flat metric?
• #### Problem 5.2.

Can you relate the flat and hyperbolic geometry of a given surface? That is, show how to uniformize the flat structure. For instance, describe (up to quasi-isometry) the hyperbolic structure on the Loch Ness monster corresponding to the flat structure given by the Chamanara surface.
Compute the rate of shortest pants decomposition.

Calculate the growth rate of balls.
• #### Problem 5.3.

In which cases do we have a Veech dichotomy on infinite translation surfaces? Are there any cases at all?
We know that there are cases in which the Veech dichotomy does not hold.
• #### Problem 5.4.

Which Veech groups arise from translation structures on the ladder surface?
For the Loch Ness Monster and for blooming Cantor tree, there are constructions to obtain Veech groups. But what about when $1<|\operatorname{Ends}_g(S)|<\infty$?
• #### Problem 5.5.

Is the billiard flow on the triangle with side lengths $3$, $4$, and $5$ ergodic?
• #### Problem 5.6.

What conditions can we put on an infinite translation surface to ensure that it contains a closed geodesic?
This problem has already been open for a while.

Cite this as: AimPL: Surfaces of infinite type, available at http://aimpl.org/genusinfinity.