Critical Exponent Rigidity for $Θ-$positive Representations
Zhufeng Yao
TL;DR
This work establishes a universal upper bound δ^α_ρ(Γ) ≤ 1 for the φ-Poincaré critical exponents of Θ-positive representations from non-elementary geometrically finite groups into real semisimple Lie groups, with equality precisely when Γ is a lattice. The authors develop a robust framework of Θ-positivity, adapted representations, and regular distortion to extend shadow and Patterson–Sullivan type methods beyond the classical PSL(2,ℝ) setting, enabling a unified treatment of Hitchin, maximal, and general Θ-positive representations. They prove ergodic and dimensional statements for the associated limit maps, including rectifiability of limit curves and bounds on the Hausdorff dimension of conical limit sets, with sharper bounds for boundary roots that involve the real rank of a Levi factor. A doubling construction and a systematic use of inner/outer radius controls underpin the rigidity results, making the techniques widely applicable to higher-rank positive representations and their dynamical footprints. Overall, the paper advances the understanding of critical exponent rigidity, ergodicity, and dimension in the Θ-positivity landscape and provides tools that may be leveraged in broader higher Teichmüller theory contexts.
Abstract
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.