Topology of domains of discontinuity for Anosov representations via circle actions

Among the remarkable properties shared with convex cocompact representations, Anosov representations admit cocompact domains of discontinuity in flag varieties. For representations produced by embedding Fuchsian representations into higher rank Lie groups, these domains are known to admit fiber bundle structures and the structure group is $\operatorname{SO}(2)$. In this article, we determine the equivariant diffeomorphism type of the fiber for these bundles when the domain lives inside a $3$-dimensional complex flag variety. In order to do so, we explicitly work out a smooth version of Fintushel's classification theorem for smooth $S^1$-actions on $4$-manifolds. We show that, in each case, the action on the fiber is equivalent to a circle action on a Hirzebruch surface (or an equivariant connected sum of such actions).