Bendings by finitely additive transverse cocycles (Q2928222)

Let \(S\) be a closed hyperbolic surface and let \(\lambda\) be a maximal geodesic lamination on \(S\). A pleated surface with pleating locus \(\lambda\) is an immersion (the ``pleating or bending map'') \(\tilde f: \mathbb H^2 \to \mathbb H^3\) between hyperbolic spaces which conjugates the universal covering group of \(S\) into a subgroup of the isometry group of \(\mathbb H^3\), and is totally geodesic on each component of \(\mathbb H^2 - \tilde \lambda\) (an ideal hyperbolic triangle) and on each geodesic of the lift \(\tilde \lambda\) of \(\lambda\) to the universal covering \(\mathbb H^2\) of \(S\). The amount of bending of a pleated surface homeomorphic to \(S\) with pleating locus \(\lambda\) is determined by a finitely additive \((\mathbb R/2\pi\mathbb Z)\)-valued transverse cocycle to the geodesic lamination, and the main result of the present paper gives a sufficient condition for such a transverse cocycle such that the pleating map \(\tilde f\) induces an injective map of the boundaries at infinity of the hyperbolic spaces (or that the pleating map induces a quasi-Fuchsian representation of the universal covering group of \(S\)). As the author notes, natural examples of finitely additive real-valued transverse cocycles arise from real Fenchel-Nielsen coordinates on pants decompositions of closed surfaces; in their proof of the surface subgroup conjecture for closed hyperbolic 3-manifolds, \textit{J. Kahn} and \textit{V. Markovic} [Ann. Math. (2) 175, No. 3, 1127--1190 (2012; Zbl 1254.57014)] used a sufficient condition on the complex Fenchel-Nielsen coordinates to obtain embedded pleated surfaces, and the main objective of the present paper is to extend this and some other previous results to arbitrary maximal geodesic laminations.