Critical Exponent Rigidity for $\hat{I}˜-$positive Representations

Abstract

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

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