3. Teichmüller theory and other tools

In the finitetype setting, a mapping class has a representative that’s a hyperbolic isometry for some hyperbolic metric exactly when it is periodic.
Problem 3.1.
Characterize the big mapping classes that can be realized by hyperbolic isometries of some hyperbolic metric on the surface. 
Let $R$ be a hyperbolic metric on $S$ and $\operatorname{Mod}(R)$ the Teichmüller modular group of $R$.
Problem 3.2.
If $f \in \operatorname{Map}(S)$ satisfies that there exists $\lambda$ such that $\forall \alpha, \beta$ and $n \in \mathbb{N}$, $i(f^n(\alpha),\beta)< c \cdot \lambda^n$, is it true then that $f \in \operatorname{Mod}(R)$ for some $R$? 
Problem 3.3.
Is there a natural bordification/boundary of Teichmüller space for big surfaces?
Remark. What kinds of properties would be desirable in a boundary? For instance, northsouth dynamics on the boundary for pseudoAnosovs.


Problem 3.4.
Are there general methods to take limits, for going from finitetype to infinitetype. What properties hold under limits? 
Problem 3.5.
Consider a geometric invariant for which we know the asymptotics in genus. This describes the generic shape of a surface of high genus. Can we use this information to obtain information about the generic shape of an infinitegenus surface? 
Come up with reasonable counting problems and analogues of geodesic currents.
Problem 3.6.
A possible counting problem is as follows. Fix a hyperbolic metric $R$ on the blooming $3$–pod surface. Fix a curve $\alpha$ and a basepoint $x \in R$, then does there exists $p(L)$ such that \[ \frac{  \phi(\alpha) : \ell_R(\phi(\alpha)) < L  }{p(L)} \simeq \text{Vol}(B_L(x))? \] 
Problem 3.7.
Give a complete quasiisometry classification (in the sense of Rosendal) of big mapping class groups. For instance, let $S$ be the ladder surface and let $S'$ be the $3$–pod blooming surface. Is $\operatorname{Map}(S)$ quasiisometric to $\operatorname{Map}(S')$?
Cite this as: AimPL: Surfaces of infinite type, available at http://aimpl.org/genusinfinity.