Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces

François Guéritaud, Fanny Kassel and Maxime Wolff

Vol. 275 (2015), No. 2, 325–359

DOI: 10.2140/pjm.2015.275.325

Abstract

We prove that any non-Fuchsian representation $\rho$ of a surface group into $PSL\left(2,ℝ\right)$ is the holonomy of a folded hyperbolic structure on the surface, unless the image of $\rho$ is virtually abelian. Using this idea, we establish that any non-Fuchsian representation $\rho$ is strictly dominated by some Fuchsian representation $j$, in the sense that the hyperbolic translation lengths for $j$ are uniformly larger than for $\rho$. Conversely, any Fuchsian representation $j$ strictly dominates some non-Fuchsian representation $\rho$, whose Euler class can be prescribed. This has applications to the theory of compact anti-de Sitter $3$-manifolds.

Keywords

representations of surface groups, folded hyperbolic structures, anti-de Sitter 3-manifolds

Mathematical Subject Classification 2010

Primary: 20H10, 32G15, 53C50

Milestones

Received: 16 November 2013

Revised: 7 July 2014

Accepted: 3 September 2014

Published: 15 May 2015