Bendings by finitely additive transverse cocycles

TL;DR

This paper characterizes how finitely additive transverse cocycles determine bending in hyperbolic surfaces and provides conditions under which these lead to quasiFuchsian representations, independent of genus.

Contribution

It introduces a genus-independent sufficient condition on finitely additive cocycles that ensures the associated pleating map yields a quasiFuchsian representation.

Findings

A complete description of bending via finitely additive cocycles.

A genus-independent criterion for quasiFuchsian representations.

Extension of pleated surface theory to finitely additive cocycles.

Abstract

Let $S$ be any closed hyperbolic surface and let $\lambda$ be a maximal geodesic lamination on $S$. The amount of bending of an abstract pleated surface (homeomorphic to $S$) with the pleating locus $\lambda$ is completely determined by an $(\mathbb{R}/2\pi\mathbb{Z})$-valued finitely additive transverse cocycle $\beta$ to the geodesic lamination $\lambda$. We give a sufficient condition on $\beta$ such that the corresponding pleating map $\tilde{f}_{\beta}:\mathbb{H}^2\to\mathbb{H}^3$ induces a quasiFuchsian representation of the surface group $\pi_1(S)$. Our condition is genus independent.