Critical Exponent Rigidity for $Θ-$positive Representations

We prove for a $Θ-$positive representation from a discrete subgroup $Γ\subset \mathsf{PSL}(2,\mathbb{R})$, the critical exponent for any $α\in Θ$ is not greater than one. When $Γ$ is geometrically finite, the equality holds if and only if $Γ$ is a lattice.

💡 Research Summary

The paper investigates the critical exponent rigidity phenomenon for Θ‑positive representations of non‑elementary discrete subgroups Γ of PSL(2,ℝ) into a simple Lie group G. A Θ‑positive representation is one that admits a continuous, ρ‑equivariant map from the limit set Λ(Γ) ⊂ S¹ to the Θ‑flag manifold F_Θ, sending every cyclically ordered n‑tuple to a Θ‑positive configuration. The authors first prove that any Θ‑positive representation is automatically Θ‑transverse, which guarantees the existence and uniqueness of a limit map (Proposition 3.19).

Using the limit map’s monotonicity, they establish precise inner and outer radius estimates for shadows of hyperbolic balls, extending the techniques of Canary‑Zhang‑Zimmer beyond the (1,1,2)‑hyper‑transverse setting. These estimates allow control of the φ‑Poincaré series Q_φ,ρ(Γ)(s)=∑_{γ∈Γ} e^{-s φ(κ(ρ(γ)))} for any linear functional φ that is a positive combination of simple roots in Θ. Consequently, for each simple root α∈Θ the associated critical exponent δ_α(ρ(Γ)) satisfies δ_α ≤ 1 (Theorem 1.2(1)).

When Γ is a lattice, the classical comparison between hyperbolic distance and Cartan projection (inequality (1.1)) yields δ_α = 1 for all α∈Θ (Theorem 1.2(2)). If Γ is geometrically finite but not a lattice, the same comparison together with Patterson–Sullivan theory shows δ_α < 1 (Theorem 1.2(3)). Thus the bound 1 is sharp and characterises lattices among geometrically finite subgroups.

The authors also prove a more general bound for any positive linear combination φ∈H⁺ (Theorem 1.3): δ_φ(ρ(Γ)) ≤ a(φ), where a(φ) is the coefficient sum of φ expressed in the basis of simple roots. Equality forces Γ to be a lattice.

In the ergodic direction, for each α∈Θ the limit map ξ_α defines a rectifiable curve ξ_α(Λ(Γ)) in the α‑flag manifold, carrying a natural Lebesgue measure m_α. Theorem 1.4 shows that when Γ is a lattice, the dynamical system (ρ(Γ), m_α) has at most D(g,α) ergodic components, where D(g,α) depends only on the Lie algebra g and the root α. Example 1.5 demonstrates that this bound is optimal for the long root in Sp(4,ℝ).

Finally, the paper relates critical exponents to the Hausdorff dimension of conical limit sets. For non‑boundary roots α∈Θ one has δ_α = dim ξ_α(Λ_c(Γ)) (Theorem 1.8(1)). For boundary roots, inequalities involving the fundamental weight ω′_α and the real rank r of the complementary semisimple part are obtained (Theorem 1.8(2)). If the representation is transverse with respect to both Θ and all adjacent roots, equality holds for every α∈Θ (Theorem 1.8(3)).

Overall, the work extends rigidity results previously known for Hitchin and maximal representations to the full class of Θ‑positive representations, providing a unified framework that links positivity, Anosov dynamics, critical exponents, and geometric properties of limit sets for a broad family of discrete subgroups of PSL(2,ℝ).