Critical Exponent Rigidity for $\Theta-$positive Representations

TL;DR

This paper establishes a rigidity property for $ heta$-positive representations, showing that the critical exponent is bounded above by one, with equality characterizing lattices among geometrically finite groups.

Contribution

It proves a new critical exponent rigidity result for $ heta$-positive representations of discrete subgroups of $ ext{PSL}(2,R)$, linking geometric finiteness and lattice conditions.

Findings

Critical exponent for $ heta$-positive representations is at most one.

Equality of critical exponent and one characterizes lattices.

Applicable to geometrically finite groups.

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.