| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

2. Big mapping class groups acting on complexes


    1. Add remark

      Problem 2.05.

      Give a classification of big mapping classes by understanding their dynamics on some combinatorial complex associated with the surface.


        • Add remark

          Problem 2.1.

          What combinatorial objects are "good" analogues of the curve complex, either uniformly for all big surfaces or for some class of big surfaces? Here "good" means that there exist relationships between topological properties of the mapping class and dynamical properties of its action on the combinatorial object.


            • Add remark

              Problem 2.15.

              Classify surfaces whose mapping class groups act continuously by unbounded orbits on a (hyperbolic) connected locally finite complex.
                  Mann-Rafi have shown that the following surfaces have trivial quasi-isometry type (in the sense of Rosendal), and thus have no continuous $\operatorname{Map}(S)$ action on a connected complex with infinite orbits: Loch Ness Monster, Cantor tree, blooming Cantor tree.


                • Add remark

                  Problem 2.2.

                  Among the surfaces which do have a hyperbolic complex on which their mapping classes act, classify the loxodromic elements. Or, more simply, give explicit constructions of loxodromic elements.


                    • Add remark

                      Problem 2.25.

                      For $S$ with no punctures and infinite genus, such as the Loch Ness Monster or the ladder surface, is there a hyperbolic complex on which $\operatorname{Map}(S)$ acts? Note that when $S$ has at least one isolated puncture, there are hyperbolic complexes on which subgroups of $\operatorname{Map}(S)$ act, but even in this case these are not obvious complexes that are invariant under all of $\operatorname{Map}(S)$. When $S$ does have at least one isolated puncture, what does the subgroup of $\operatorname{Map}(S)$ that stabilizes a single isolated puncture or finitely many isolated punctures look like inside $\operatorname{Map}(S)$?


                        • Add remark

                          Problem 2.3.

                          Classify surfaces with finite-invariance index 1, 2, and 3.

                          (Note: finite invariance index is defined by Durham-Fanoni-Vlamis.)


                            • Add remark

                              Problem 2.35.

                              Is there a good graph for surfaces with finite-invariance index 0?
                                  If the action of the mapping class group on the graph is required to be continuous with unbounded orbits, recent work of Mann-Rafi implies that the answer is no since these surfaces will have CB mapping class groups.


                                • Add remark

                                  Problem 2.4.

                                  When a given surface has multiple hyperbolic complexes associated to it, what relationships exist between those complexes? Are they quasi-isometric? Are there Lipschitz maps between them? For instance, the relative arc graph (also called the loop graph) and the ray graph are quasi-isometric. What else?


                                    • Add remark

                                      Problem 2.45.

                                      For a Cantor set $C$, $\operatorname{Map}(\mathbb{R}^2- C)$ is not acylindrically hyperbolic, and so it contains no WPDs. Are there any WWPDs elements? If so, describe them.
                                          This question has since been answered by Rasmussen; he gives a characterization of WWPD elements.


                                        • Add remark

                                          Problem 2.5.

                                          When a graph associated to a surface is hyperbolic, what can be said about its Gromov boundary? For instance, can we describe the boundary of the ray/loop graph as space of laminations?


                                            • Add remark

                                              Problem 2.55.

                                              Construct mapping classes that act parabolically on the ray graph of $\mathbb{R}^2-C$, where $C$ is a cantor set.


                                                • Add remark

                                                  Problem 2.6.

                                                  Consider the horseshoe map. After puncturing along its invariant Cantor set, is the resulting mapping class loxodromic on the ray graph? What about other "naturally occurring" homeomorphisms?

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