
## 3. Teichmüller theory and other tools

1.     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 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?
There are at least two answers that address this question in the recent work of Saric.
1. Remark. What kinds of properties would be desirable in a boundary? For instance, north-south dynamics on the boundary for pseudo-Anosovs.
• #### Problem 3.4.

Are there general methods to take limits, for going from finite-type to infinite-type. What properties hold under limits?
When $S$ has finite genus, there is an inverse limit structure and some properties for the pure mapping class group, like residual finiteness, can be inherited from finite-type surfaces.
• #### 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 infinite-genus 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 quasi-isometry 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 $\operatorname{Map}(S)$ quasi-isometric to $\operatorname{Map}(S')$?

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