Critical Exponent Rigidity for $\Theta -$positive Representations

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.

arXiv commentVer 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