We consider hyperbolic and anti-de Sitter (AdS) structures on MΓ(0,1), where M is a d-dimensional Gromov-Thurston manifold. If M has
cone angles greater than 2Ο, 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 2Ο,
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 2kβ3.
When d=3, the moduli space of quasifuchsian AdS (resp. hyperbolic)
manifolds diffeomorphic to MΓ(0,1) contains a submanifold of dimension
2kβ2, 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}.Comment: 48 page