Hausdorff Dimension of Anosov Subgroups' Limit Sets with Special Self-Affine Complexity

Abstract: Let $Γ\subset \mathsf{PGL}(d,\mathbb{R})$ be an irreducible projective Anosov subgroup and let $Λ<sup>1(Γ)$ be its projective limit set. Viewing $Λ<sup>1(Γ)$ as an analogue of a self-affine set, we investigate the Hausdorff dimension of $Λ<sup>1(Γ)$ under specific assumptions regarding its affine complexity: 1. If $Λ<sup>1(Γ)$ is of full Hausdorff dimension, then $d= 2$ and $Γ$ is a cocompact lattice. 2. If $d = 3$ and $Γ$ is the image of a closed surface group under an irreducible Anosov representation, then $Λ<sup>1(Γ)$ never has Hausdorff dimension $1$ unless the representation is Hitchin. 3. If the limit set $Λ<sup>1(Γ)$ exhibits a partial quasi-self-similarity (in the sense of Falconer~\cite{falconerselfsimilar1}) -- which can be implied by the ``regular distortion property'' of $Γ$ -- then the Hausdorff dimension of $Λ<sup>1(Γ)$ equals the critical exponent of the first simple root. An application of this result is the computation of the Hausdorff dimension of the limit set for arbitrary $Θ$-positive representations of convex cocompact Fuchsian groups.

• The paper establishes critical exponent rigidity by showing that for each simple root, the critical exponent is at most 1, reaching equality only for lattice groups.
• It precisely links the Hausdorff dimension of limit sets to the critical exponent, demonstrating a direct correspondence especially for non-boundary roots.
• The study employs cone dynamics, approximate additivity of Poincaré series, and robust shadow lemma techniques to derive sharp bounds and ergodicity results.

Hausdorff Dimension of Anosov Subgroups' Limit Sets with Special Self-Affine Complexity

Introduction and Background

The paper investigates the metric and dynamical invariants associated with $\Theta$-positive representations of discrete subgroups $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$ into higher-rank simple Lie groups $G$, focusing on the Hausdorff dimension of their limit sets and critical exponent rigidity. This work sits at the intersection of higher Teichmüller theory, dynamical systems, and the geometry of discrete subgroups of Lie groups.

In higher Teichmüller theory, positivity and Anosov properties underpin the classification and analysis of well-behaved representations. Hitchin representations and maximal representations serve as principal examples, both known to be Anosov and admit equivariant positive boundary maps. Recent advances—particularly the $\Theta$-positivity framework of Guichard–Wienhard—generalize the notions of positivity to a larger class of semisimple Lie groups and roots, yielding new families of higher Teichmüller spaces and representations with rich geometric and dynamical structures.

Main Results

The central results establish critical exponent rigidity for $\Theta$-positive representations and relate the Hausdorff dimension of limit sets to algebraic invariants associated with the representation. Primary findings include:

(1) Critical Exponent Upper Bound and Rigidity

Let $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$ be a non-elementary discrete subgroup, $G$ a simple real Lie group with $\Theta$-positive structure, and $\rho:\Gamma \to G$ a $\Theta$-positive representation. For each simple root $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$0, defining the associated Poincaré series and critical exponent $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$1, the following rigidity holds:

• $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$2 for every $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$3.
• If $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$4 is a lattice, then equality holds: $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$5.
• If $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$6 is geometrically finite but not a lattice, then strict inequality: $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$7.

Moreover, for any positive functional $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$8, $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$9, with equality only in the lattice case.

(2) Hausdorff Dimension and Limit Sets

For conical limit sets $G$0 and projection of limit maps to flag manifolds $G$1, the paper establishes:

• For non-boundary simple roots $G$2, the Hausdorff dimension equals the critical exponent: $G$3.
• For boundary roots $G$4, sharp inequalities relate the dimension to the critical exponent and the fundamental weights for Levi subgroups associated to $G$5. In particular, the lower bound involves the real rank $G$6 and the symmetrized root functional, improving upon previous bounds in the literature.
• When the representation is also transverse with respect to adjacent roots, equality holds for all $G$7.

These estimates hold uniformly across a broad class of Lie groups admitting $G$8-positivity, not restricted to algebraic or split real forms.

(3) Measure-theoretic Ergodicity of Limit Sets

When $G$9 is a lattice and $\Theta$0 is $\Theta$1-positive:

• For each $\Theta$2, the action of $\Theta$3 on the limit set $\Theta$4 (with respect to the Hausdorff/Lebesgue measure) has at most $\Theta$5 ergodic components. For non-boundary roots, the measure is ergodic; for boundary roots, the bound can be strictly higher, and explicit examples show optimality.

Techniques and Proof Ideas

The authors' strategy combines:

• Cone dynamics and positivity: The structure of $\Theta$6-positive semigroups and cones leads to monotonicity and distortion control, leveraging differentiable chart systems adapted to the cone geometry.
• Approximate additivity and Poincaré series: The strong coarse additivity of Cartan projections (established via limit map properties and the adapted representation) enables analogues of classical subadditivity arguments, crucial for determining critical exponents and relating them to metric properties.
• Shadow lemma machinery: A central technical ingredient is a robust shadow lemma for $\Theta$7-positive representations, ensuring that the mass of a shadow in the limit set is tightly controlled by the group elements’ Cartan projections. This is essential for estimating the Hausdorff dimension from below.
• Doubling constructions: The doubling of $\Theta$8 (when it is not a lattice) together with doubling of the representation allows the passage between geometrically finite and lattice settings, showing that the strict entropy drop applies unless $\Theta$9 is a lattice.
• Ergodicity arguments: The rank of the tangent cone at the identity to the positive semigroup directly bounds the number of ergodic components of the Hausdorff measure, providing sharp examples for boundary roots.

Notable and Contradictory Claims

• The upper bound for all simple root exponents at unity generalizes previously known results for Hitchin and maximal representations, extending them to higher rank and more general Lie groups admitting $\Theta$0-positivity.
• The direct equality between Hausdorff dimension and the critical exponent for non-boundary roots claims a tight link between the geometry of the limit set and the representation-theoretic data, provided certain transversality properties hold.
• For boundary roots, the lower bound incorporates an explicit factor of real rank, giving a strictly better estimate than those available from older techniques (notably in non-split or tube type settings).

Implications and Future Directions

Theoretical Implications

This work substantially bridges representation-theoretic positivity notions with fine quantitative geometry and dynamics of limit sets. The extension of dimension estimates and critical exponent rigidity from classical settings (e.g., Fuchsian and Hitchin groups) to general $\Theta$1-positive structures demonstrates the universality and robustness of these techniques.

Moreover, the measure-theoretic classification of ergodic components provides a new lens to analyze rigidity: the rank of the underlying cone detects nontrivial decompositions, potentially relating to symmetry breaking and the failure of uniqueness in boundary map projections for boundary roots.

Practical Relevance

The results may impact the study of the dynamics and geometry of discrete subgroups in homogeneous spaces and their moduli, particularly in real projective and pseudo-Riemannian contexts where Anosov and positivity properties ensure properness and geometric finiteness.

Future Developments

• Strong rigidity and Zariski closure: The paper highlights the open question of whether the characterization of equality in the critical exponent (i.e., images lying in irreducible copies of $\Theta$2) extends to general $\Theta$3-positive representations, potentially linking to Zariski density.
• Sharpness and boundary root regularity: Determining the exact dimensional relation for boundary roots in all cases, and potentially constructing counterexamples or verifying the sharpness of bounds, remains an avenue for further exploration.
• Broader applicability: The regular distortion and shadow lemmas developed may contribute to the analysis of limit sets and dimensions in broader classes of Anosov groups, even beyond those admitting positivity, including applications to convex projective and higher-rank locally symmetric spaces.

Conclusion

This paper rigorously establishes upper bounds and rigidity statements for the critical exponents associated with $\Theta$4-positive representations, links the Hausdorff dimension of the limit sets with algebraic and dynamical data of the representation, and quantifies measure-theoretic ergodicity in terms of cone rank. These results generalize, unify, and sharpen prior theorems in higher Teichmüller theory, with strong implications for the geometry and dynamics of discrete subgroups in higher-rank Lie groups. The methods developed offer a template for further advances in the quantitative theory of discrete group actions on flag varieties and homogeneous spaces.

Reference: "Hausdorff Dimension of Anosov Subgroups' Limit Sets with Special Self-Affine Complexity" (2604.18365).