
## 1. Classification of elements of big mapping class groups

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.

Give a Nielsen-Thurston classification type theorem for big mapping classes.
The following definition was proposed during the discussion on open problems.

Suppose a big mapping class is reducible if it preserves a (possibly infinite) discrete collection of pairwise disjoint simple closed curves and simple proper arcs (where discrete means that the collection "does not accumulate anywhere inside the surface"), and irreducible otherwise.
• #### Problem 1.2.

Does every big mapping class either preserve a hyperbolic metric or preserve a lamination?
•     This was proposed as an "easier version" of the previous problem.

#### Problem 1.3.

Given an (irreducible) big mapping class $f$ and a simple closed curve $\alpha$, do $f^n(\alpha)$ and $f^{-n}(\alpha)$ converge to a lamination?
• #### Problem 1.4.

Describe all big mapping classes that preserve a train track on the surface.
• #### Problem 1.5.

Is there a dynamical description of an irreducible mapping class? For instance, given two simple closed curves $\alpha$ and $\beta$, what can be said about the asymptotics of $i(f^n(\alpha,\beta))$?
For compact surfaces, this grows exponentially. We have produced examples on big surfaces with exponential growth with polynomial decay. What else?
• #### Problem 1.6.

Given a pseudo-Anosov acting on a translation surface, what can be said about the measured laminations in the stable and in the unstable direction? In general, "How much of Thurston’s notes goes through?"
• #### Problem 1.7.

Can we characterize big mapping classes whose mapping tori admit a complete hyperbolic metric?

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