Gromov-Thurston manifolds and anti-de Sitter geometry

Abstract

We consider hyperbolic and anti-de Sitter (AdS) structures on $M\times (0,1)$, where $M$ is a $d$-dimensional Gromov-Thurston manifold. If $M$ has cone angles greater than $2\pi$, we show that there exists a "quasifuchsian" (globally hyperbolic maximal) AdS manifold such that the future boundary of the convex core is isometric to $M$. When $M$ has cone angles less than $2\pi$, there exists a hyperbolic end with boundary a concave pleated surface isometric to $M$. Moreover, in both cases, if $M$ is a Gromov-Thurston manifold with $2k$ pieces (as defined below), the moduli space of quasifuchsian AdS structures (resp. hyperbolic ends) satisfying this condition contains a submanifold of dimension $2k-3$. When $d=3$, the moduli space of quasifuchsian AdS (resp. hyperbolic) manifolds diffeomorphic to $M\times (0,1)$ contains a submanifold of dimension $2k-2$, and extends up to a "Fuchsian" manifold, that is, an AdS (resp. hyperbolic) warped product of a closed hyperbolic manifold by~$\R$. We use this construction of quasifuchsian AdS manifolds to obtain new compact quotients of $\O(2d,2)/\U(d,1)$. The construction uses an explicit correspondence between quasifuchsian $2d+1$-dimensional AdS manifolds and compact quotients of $\O(2d,2)/\U(d,1)$ which we interpret as the space of timelike geodesic Killing fields of $\AdS^{2d+1}$.