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1. Higher rank abelian group actions

    1. Totally Anosov TNS action


      Add remark

      Problem 1.1.

      [B. Kalinin] Let $\alpha$ be a smooth action on a closed manifold $X$ by $\mathbb{Z}^{k}$, $k\ge 2$. Assume that every elements in $\mathbb{Z}^{k}$ is an Anosov diffeomorphism and the action is totally non symplectic (TNS), that is there are no negatively proportional Lyapunov exponents. Is $X$ diffeomorphic to a nilmanifold and $\alpha$ smoothly conjugate to an affine action?
        1. Remark. Note that TNS implies $k\ge 2$. Furthermore, TNS is stronger than no rank one factor assumption.

            • Classification of Affine Anosov action


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              Problem 1.2.

              [K. Vinhage] Let $G=\mathbb{Z}^{k}\times \mathbb{R}^{l}$, $k+l\ge 2$.
              1. Which affine actions of $G$ are Anosov?
              2. Are there any non-standard ones?
              3. Which ones have rank one factor?
              4. Which ones contain ergodic, partially hyperbolic (but not Anosov) elements?
                1. Remark. [K Vinhage] 1. Here, by affine *actions*, we mean the composition of a translation and automorphism on a double homogeneous space.

                  2. Our understanding of algebraic Anosov actions has grown, but usually only for certain specific families of examples. These were called "standard" by Katok and Spatzier (Section 2.2 in [MR1307298]). If we knew this list was exhaustive, we would be able to claim local rigidity for any algebraic Anosov action.

                  3. Rank one factors are obstructions to rigidity. These can exist in a variety of categories. It would be nice to know, for instance, that if there exists a measurable rank one factor, then there is an affine rank one factor, and how to detect this by looking at the algebraic data.

                  4. The only known example of this that I know is direct products of Anosov flows and diffeomorphisms (at least, after passing to suspensions).

                    • Perturbation of rank one factor

                          Rank one factors are generally considered to be obstructions to rigidity as they can be perturbed. Answering this question would solidify this perspective, since it would show that the original action could also be perturbed once the base is. This is obvious for direct products, but not clear at all for fibered transformations.

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                      Problem 1.3.

                      [K. Vinhage] Let $\alpha$ be a $C^{\infty}$ $\mathbb{R}^{k}$ action on a manifold $X$ with $C^{\infty}$ rank one factor $\mathbb{R}$ action $\beta$ on $Y$. Does any perturbation of $\beta$ lift to a perturbation of $\alpha$?

                        • Perturbation of product action


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                          Problem 1.4.

                          [A. Wilkinson] Let $G=\mathbb{Z}^{k}$ or $\mathbb{R}^{k}$ with $k\ge 2$. Assume that $G$ action $\alpha$ on a closed manifold $X$ is Anosov. Fix a closed manifold $Y$. When is every perturbation of $\alpha\times Id$ on $X\times Y$ is a smooth product?


                            • Add remark

                              Problem 1.5.

                              [K. Vinhage] Suppose $\mathbb{R}^{2}$ acts smoothly on a closed manifold $X$ with invariant foliation $F$. Assume that
                              1. The set of normally hyperbolic elements is dense in $\mathbb{R}^{2}$.
                              2. It satisfies accessibility property.
                              3. Every coarse Lyapunov subspaces is one dimensional.
                              4. Every element is transitive.
                              Then is the action is algebraic?
                                1. Remark. [K. Vinhage] These assumptions are generalizations of the totally Cartan and no-rank-one-factor assumptions in the work of Spatzier-Vinhage ([arXiv:1901.06559]) to a partially hyperbolic setting. The models for these actions are restrictions of accessible totally Cartan $\mathbb{R}^k$ actions to a generic $\mathbb{R}^2$ subgroup. Local rigidity of several such algebraic actions is known. This would be an extension of the local rigidity phenomenon to give a complete classification.

                                      Cite this as: AimPL: Global rigidity of actions by higher-rank groups, available at http://aimpl.org/rigidhigherrank.