Gromov–Thurston manifolds and anti-de Sittergeometry

We consider hyperbolic and anti-de Sitter (AdS) structures on M .0;1/, where M is a d -dimensional Gromov-Thurston manifold.If M has cone angles greater than 2 , 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 , 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 D 3, the moduli space of quasifuchsian AdS (resp.hyperbolic) manifolds diffeomorphic to M .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 then 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 .2dC1/dimensionalAdS 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 C1 .