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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.