2023/10/18 by Monclair, Daniel, Schlenker, Jean-Marc, Tholozan, Nicolas · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2310.12003
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 2π, 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 2π, 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× (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 Ø(2d,2)/\U(d,1). The construction uses an explicit correspondence between quasifuchsian 2d+1-dimensional AdS manifolds and compact quotients of Ø(2d,2)/\U(d,1) which we interpret as the space of timelike geodesic Killing fields of \AdS2d+1.