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1. Dyamics for special subgroups of the mapping class group

    1. Torelli group orbits

          The Torelli group $T_{g,n}$ is the kernel of the action of the mapping class group of $\Sigma_{g,n}$ on homology $H_1(\Sigma_{g,n}, \Z)$. Restricting the mapping class group action to this subgroup gives an action on the character variety $M(\Sigma_{g,n}, r)$ of rank $r$ semi-simple local systems.

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      Problem 1.1.

      Classify the finite orbits of $T_{g,n}$ on the character varieties $M(\Sigma_{g,n}, r)$.
          For $r = 1$ the Torelli group acts trivially. Some possible interesting examples: representations valued in the infinite Dihedral group inside $\mathrm{SL}_2$.

        • Mapping class group orbits in higher rank


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          Problem 1.2.

          What are some sufficient conditions for a point to have a Zariski dense orbit under the mapping class group on $M_\gamma(\Sigma_{g,n}, \SL_3)$? Here $\gamma$ is a fixed characteristic polynomial for monodromy around the punctures.

            • Classification of MCG-finite localy systems of higher rank

                  Recent work of Lam, Landesman, Litt has led to a mostly complete understanding of the finite orbits under the mapping class group of rank $2$ local systems on $\Sigma_{0,n}$.

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              Problem 1.3.

              Can we classify the finite orbts under the mapping class group in $M(\Sigma_{0,n}, \mathrm{SL}_3)$, in particular the case $n = 4$?

                • Fixed points of special elements of the mapping class group


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                  Problem 1.4.

                  Fix $\gamma \in \mathrm{Mod}(\Sigma_g)$ can we understand the set of fixed points $M(\Sigma_g, 2)(\overline{\mathbb{Z}})^{\gamma}$ of its action on the rank $2$ character variety?

                      Cite this as: AimPL: Motives and mapping class groups, available at http://aimpl.org/motivesandmcg.