Critical Exponent Rigidity for $\Theta-$positive Representations

Zhufeng Yao

Critical Exponent Rigidity for $\Theta-$positive Representations

Differential Geometry 2026-02-09 v4 Dynamical Systems Group Theory

Authors: Zhufeng Yao

Abstract

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

Keywords

group theory theta correspondence

Cite

@article{arxiv.2505.17559,
  title  = {Critical Exponent Rigidity for $\Theta-$positive Representations},
  author = {Zhufeng Yao},
  journal= {arXiv preprint arXiv:2505.17559},
  year   = {2026}
}

Comments

Ver 4: 67 pages, with corrections on typos and errors. And several proofs have improved, especially the whole Section 4.3 and the proof of Proposition 6.1

Critical Exponent Rigidity for $\Theta-$positive Representations

Abstract

Keywords

Cite

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