Topology of domains of discontinuity for Anosov representations via circle actions

TL;DR

The paper studies cocompact domains of discontinuity for θ-Anosov representations into complex classical groups, focusing on iota-Fuchsian representations into 3-dimensional flag varieties. It reduces the topological question to the fiber topology of an SL(2, R)-equivariant fibration over a hyperbolic surface, and then determines the fiber diffeomorphism type by adapting Fintushel’s smooth S^1-action classification via tangential weight graphs. The key technical advance is identifying the fiber with a Hirzebruch surface (or a connected sum of such) and describing the SO(2) action as an algebraic action on these surfaces, in several explicit cases (Flag(C^3), CP^3, Lag(C^4)). The approach leverages a smooth variant of Fintushel’s circle-action classification, the Kapovich–Leeb–Porti framework for domains of discontinuity via balanced ideals, and detailed weight-data analysis on flag varieties to connect representation theory with four-manifold topology. These results illuminate the topology of Anosov-domain quotients in higher rank and provide a concrete geometric model for the fiber in terms of well-understood complex surfaces, with potential implications for the structure of Anosov components in character varieties.

Abstract

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).

Figures (14)

• Figure 1: Hasse diagram of Bruhat order on $W=S_3$
• Figure 2: Bruhat order on $W_{\Delta,\{2\}}$ for $G=\mathop{\mathrm{Sp}}\nolimits(4,\mathbb{C})$
• Figure 3: Lift of Fuchsian representation to $\mathop{\mathrm{SL}}\nolimits(2,\mathbb{R})$
• Figure 4: Tangential weight graph for $S^1$--action on $\mathbb{CP}^1\times\mathbb{CP}^1$
• Figure 5: Equivariant connected sum of $S^1$--actions on $S^2\times S^2$

Theorems & Definitions (103)

• Example 2.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof