Bendings by finitely additive transverse cocycles

Abstract

Let $S$ be any closed hyperbolic surface and let $λ$ be a maximal geodesic lamination on $S$. The amount of bending of an abstract pleated surface (homeomorphic to $S$) with the pleating locus $λ$ is completely determined by an $(\mathbb{R}/2π\mathbb{Z})$-valued finitely additive transverse cocycle $β$ to the geodesic lamination $λ$. We give a sufficient condition on $β$ such that the corresponding pleating map $\tilde{f}_β:\mathbb{H}^2\to\mathbb{H}^3$ induces a quasiFuchsian representation of the surface group $π_1(S)$. Our condition is genus independent.