[Paper Review] Domains of discontinuity in oriented flag manifolds

Florian Stecker, Nicolaus Treib|arXiv (Cornell University)|Jun 12, 2018

Advanced Algebra and Geometry4 citations

TL;DR

This paper extends the theory of domains of discontinuity to oriented flag manifolds by constructing cocompact, properly discontinuous actions of Anosov subgroups on oriented flag manifolds $G/P_R$ via removal of sets derived from the limit set. It provides a combinatorial description using balanced ideals in the extended Weyl group and gives the first examples of such domains that are not lifts from unoriented flag manifolds, including nonempty cocompact domains in oriented Grassmannians for Hitchin representations.

ABSTRACT

We study actions of discrete subgroups $Γ$ of semi-simple Lie groups $G$ on associated oriented flag manifolds. These are quotients $G/P$, where the subgroup $P$ lies between a parabolic subgroup and its identity component. For Anosov subgroups $Γ\subset G$, we identify domains in oriented flag manifolds by removing a set obtained from the limit set of $Γ$, and give a combinatorial description of proper discontinuity and cocompactness of these domains. This generalizes analogous results of Kapovich-Leeb-Porti to the oriented setting. We give first examples of cocompact domains of discontinuity which are not lifts of domains in unoriented flag manifolds. These include in particular domains in oriented Grassmannians for Hitchin representations, which we also show to be nonempty. As a further application of the oriented setup, we give a new lower bound on the number of connected components of $B$-Anosov representations of a closed surface group into $\mathrm{SL}(n,\mathbb{R})$.

Motivation & Objective

• To generalize the construction of cocompact domains of discontinuity from unoriented to oriented flag manifolds.
• To provide a combinatorial characterization of proper discontinuity and cocompactness using balanced ideals in the extended Weyl group.
• To construct the first examples of cocompact domains of discontinuity in oriented flag manifolds that are not lifts from unoriented ones.
• To establish a new lower bound on the number of connected components of $B$–Anosov representations into $\operatorname{SL}(n,\mathbb{R})$ using the oriented framework.

Proposed method

• The authors define oriented flag manifolds $\mathcal{F}_R = G/P_R$ where $P_R$ is a parabolic subgroup containing the identity component of a parabolic $P_\theta$, generalizing unoriented flag manifolds.
• They introduce the concept of balanced ideals in the extended Weyl group $\widetilde{W}_{R,S}$ to describe the set of points to be removed from $\mathcal{F}_S$ to obtain a $\Gamma$–invariant domain $\Omega \subset \mathcal{F}_S$.
• Using the limit map $\xi: \partial_\infty \Gamma \to \mathcal{F}_\theta$, they construct a $\Gamma$–equivariant map $\mathcal{Q}: \mathcal{F}_R \to \mathcal{C}(\mathcal{F}_S)$, whose fibers define the sets to be removed.
• They prove proper discontinuity and cocompactness of the action on $\Omega$ by showing strong transverse expansion of $\Gamma$ near the removed set, leveraging the dynamics of the limit set and refined Schubert strata.
• The construction relies on the Bruhat order and transverse relative positions in the oriented setting, with special attention to the role of the opposition involution and the extended Weyl group.
• They apply the framework to Hitchin representations and generalized Schottky representations, showing nonemptiness and cocompactness of the resulting domains in oriented Grassmannians.

Experimental results

Research questions

• RQ1 Can cocompact domains of discontinuity be constructed in oriented flag manifolds for Anosov representations, beyond lifts from unoriented ones?
• RQ2 What combinatorial structure in the extended Weyl group characterizes proper discontinuity and cocompactness in the oriented setting?
• RQ3 Do Hitchin representations in $\operatorname{SL}(n,\mathbb{R})$ admit nonempty cocompact domains of discontinuity in oriented Grassmannians?
• RQ4 How does the oriented flag manifold framework yield new lower bounds on the number of connected components of $B$–Anosov representations?
• RQ5 What is the geometric significance of relative positions and refined Schubert strata in the context of oriented flag manifolds?

Key findings

• The paper constructs the first known examples of cocompact domains of discontinuity in oriented flag manifolds that are not lifts from unoriented flag manifolds.
• For Hitchin representations into $\operatorname{SL}(n,\mathbb{R})$, the resulting domains in oriented Grassmannians are nonempty, providing new geometric structures on the quotient.
• The authors establish a new lower bound on the number of connected components of $B$–Anosov representations into $\operatorname{SL}(n,\mathbb{R})$ using the oriented setup.
• The construction of domains relies on balanced ideals in the extended Weyl group $\widetilde{W}_{R,S}$, which generalize the balanced ideals used in the unoriented case.
• The action of $\Gamma$ on the domain $\Omega \subset \mathcal{F}_S$ is shown to be properly discontinuous and cocompact via transverse expansion dynamics near the limit set.
• The framework applies to both Hitchin representations and generalized Schottky representations, demonstrating broad applicability in higher-rank Anosov dynamics.