Given a smooth spacelike surface $Σ$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $ρ:π_1(S)\to\mathrm{PSL}_2\mathbb{R}\times\mathrm{PSL}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\geq 2$, a canonical construction associates to $Σ$ a diffeomorphism $ϕ_Σ$ of $S$. It turns out that $ϕ_Σ$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ on $S$ induced by the action of $ρ$ on $\mathbb{H}^2\times\mathbb{H}^2$. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that $ϕ_Σ$ is the composition of a Hamiltonian symplectomorphism of $(S,h)$ and the unique minimal Lagrangian diffeomorphism from $(S,h)$ to $(S,h')$.
On a closed symplectic surface Sigma of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group Symp(Sigma) to the cohomology group H^1(Sigma;R) that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or more, we give another new construction of an extended flux map using hyperbolic geometry.
We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the "other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these m…
Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichmüller space of crowned hyperbolic surfaces and as the bundle over the Teichmüller space of $Σ$ of meromorphic quadratic differentials with poles of order at least $3$ at the punctures.
Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of $Σ$ and as the bundle over the Teichmüller space of $Σ$ whose fibre consists of meromorphic quadratic differentials with poles of order at most $2$ at the punctures.
We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional ideal cuboctahedron. We show the existence of a similar family of collapsing anti-de Sitter polytopes, and join the two deformations by means of an opportune half-pipe orbifold structure. The desired examples of geometric transition are then obtained by gluing copies of the poly…
We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) 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 compact anti-de Sitter 3…
In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles the complex Fenchel-Nielsen coordinates for hyperbolic quasi-Fuchsian manifolds.
Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces c…
We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface $S$, in relation to some geometric invariants depending only on the two points in Teichmüller space of $S$ provided by Mess' parameterization - namely on two isotopy classes of hyperbolic metrics $h$ and $h'$ on $S$. The main result of the paper is that the volume coarsely behaves like the minima of the $L^1$-energy of maps from $(S,h)$ to $(S,h')$. The study of $L^p$-type energies had been suggested by Thurston, in contrast with the well-studied Lipschitz distance. A corol…
Let $M$ be a globally hyperbolic maximal compact $3$-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that $M$ admits a unique foliation by constant mean curvature surfaces. In this paper we extend this result to singular spacetimes with particles (cone singularities of angles less than $π$ along time-like geodesics).
We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into acylindrically hyperbolic groups has an absolutely elliptic image. This result provides a non-arithmetic generalization of homomorphism superrigidity of Farb--Kaimanovich--Masur and Bridson--Wade.
We consider globally hyperbolic maximal anti de Sitter 3-manifolds $M$ with a closed Cauchy surface $S$ of genus greater than one and prove that any pair of hyperbolic metrics on $S$ can be realized as the boundary metrics of the convex core of a maximal globally hyperbolic anti de Sitter 3-manifold structure on $M$. This answers the existence part of a question of Mess about the unique realization of such metrics. Our theorem has a nice formulation purely in terms of 2-dimensional Teichmüller theory and earthquakes.
In this note we show that any homomorphism from a pure surface braid group to a torsion-free hyperbolic group either has a cyclic image or factors through a forgetful map. This extends and gives a new proof of an earlier result of the author which works only when the target is a free group or a surface group. We also prove a similar rigidity result for the pure braid group of the disk.
We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $π$. We interpret this result in terms of Teichmüller theory, and prove the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity between two hyperbolic surfaces with cone singularities when the cone angles are the same for both surfaces and are less than $π$.
We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than $π$) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective…
Let $λ_-$ and $λ_+$ be two bounded measured laminations on the hyperbolic disk $\mathbb H^2$, which "strongly fill" (definition below). We consider the left earthquakes along $λ_-$ and $λ_+$, considered as maps from the universal Teichmüller space $\mathcal T$ to itself, and we prove that the composition of those left earthquakes has a fixed point. The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism $u:{\mathbb RP}^1\to {\mathbb RP}^1$, the boundary of the convex hull in $AdS^3$ of its graph in ${\mathbb RP}^1\times{\mathbb RP}^1\simeq \partial AdS^3$ is the disjoint …
A survey on the recent work of Danciger, Guéritaud and Kassel on Margulis space-times and complete anti-de Sitter space-times. Margulis space-times are quotients of the 3-dimensional Minkowski space by (non-abelian) free groups acting propertly discontinuously. Goldman, Labourie and Margulis have shown that they are determined by a convex co-compact hyperbolic surface $S$ along with a first-order deformation of the metric which uniformly decreases the lengths of closed geodesics. Danciger, Guéritaud and Kassel show that those space-times are principal $\mathbb R$-bundles over $S$ with time-lik…
For $d=4, 5, 6, 7, 8$, we exhibit examples of $\mathrm{AdS}^{d,1}$ strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space $\mathbb{H}^d$, nor to any symmetric space. This provides a negative answer to Question 5.2 in [9A12] and disproves Conjecture 8.11 of Barbot-Mérigot [BM12]. We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong's hyperbolicity criterion [Mou88] for Coxeter groups built on Danciger-Guéritaud-Kassel [DGK17] and find examples of Coxeter groups $W$ such that the s…
We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power…
In this short note we explain how to adapt the construction of two Riemannian metrics on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component to the deformation space of globally hyperbolic anti-de Sitter structures: the pressure metric and the Loftin metric (studied by Qiongling Li). We show that the former is degenerate and we characterize its degenerate locus, whereas the latter is nowhere degenerate and the Fuchsian locus is a totally geodesic copy of Teichmüller space endowed with a multiple of the Weil-Petersson metric.
We study homomorphisms from Kähler groups to Coxeter groups. As an application, we prove that a cocompact complex hyperbolic lattice (in complex dimension at least 2) does not embedd into a Coxeter group or a right-angled Artin group. This is in contrast with the case of real hyperbolic lattices.
We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature -1. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichmüller theory, as well as analogs in hyperbolic geometry.
We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra that generalizes the complex numbers. Equivalently, tetrahedra with lightlike faces are determined by a pair of edge lengths and their duals by a pair of dihedral angles. We prove that the dual tetrahedra are precisely the generalized ideal tetrahedra introduced by Danciger. Final…
We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these metrics has a simple description in terms of Teichmüller theory. In the hyperbolic settings both questions have positive answers for a certain subset of the quasi-Fuchsian manifolds: those containing a closed surface with principal curvatures at most 1. We show that this subset is parameterized by an open domain of the cotangent bundle of Teichmüller space. Thes…
