
## 1. Classifying big mapping classes

In the finite-type setting, the Nielsen–Thurston classification says that every mapping class is either periodic or reducible (or both), or else it is pseudo-Anosov. Can we give a classification of big mapping classes?
1. #### Problem 1.1.

[Priyam Patel] Does every big mapping class either preserve a metric or preserve a lamination?
• #### Problem 1.2.

Is there a dynamical description of all mapping classes that do not preserve any (possibly infinite) multicurve?
•     This was proposed as an "easier version" of Problem 1.2.

#### Problem 1.3.

[Kasra Rafi] Let $f$ be an (irreducible) big mapping class and $\alpha$ a simple closed curve. Does the sequence $f^n(\alpha)$ converge to a lamination?
(Jing Tao and Mladen Bestvina): But what about translations? (Kasra Rafi): Then let the union of the orbit $f^n(\alpha)$ converge to a filling system. Then does $f^n(\alpha)$ to a lamination?
• #### Problem 1.4.

[Autumn Kent] Is there a natural bordification of Teichmüller space for big surfaces?
(Carolyn Abbott): There are at least two answers that address this question in recent work of Saric. (Kasra Rafi): What kinds of properties would be desirable in a boundary? For instance, north-south dynamics on the boundary for pseudo-Anosovs.
• #### Problem 1.5.

[Priyam Patel] How should handle shifts fit into a classification scheme for mapping classes?
(Jing Tao): These preserve a metric. (Priyam Patel): What about the composition of a few handle shifts?
• #### Problem 1.6.

[Katie Mann] Describe all big mapping classes that have a train track.
•     In the finite-type setting, a mapping class has a representative that’s a hyperbolic isometry for some hyperbolic metric exactly when it is periodic.

#### Problem 1.7.

[org.aimpl.user:jlanier8@gatech.edu] Characterize the big mapping classes that can be realized by hyperbolic isometries.
• #### Problem 1.8.

[Katie Mann] Give a classification of big mapping classes by understanding their dynamics on some combinatorial complex associated with the surface.
• #### Problem 1.9.

[Nic Brody] Characterize big mapping classes whose mapping tori are hyperbolic.

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