The Universal Phase Space of AdS3 Gravity

We describe what can be called the "universal" phase space of AdS3 gravity, in which the moduli spaces of globally hyperbolic AdS spacetimes with compact spatial sections, as well as the moduli spaces of multi-black-hole spacetimes are realized as submanifolds. The universal phase space is parametrized by two copies of the Universal Teichm\"uller space T(1) and is obtained from the correspondence between maximal surfaces in AdS3 and quasisymmetric homeomorphisms of the unit circle. We also relate our parametrization to the Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the holographic (Fefferman-Graham) description. In particular, we obtain a relation between the generators of quasiconformal deformations in each T(1) sector and the chiral Brown-Henneaux vector fields. We also relate the charges arising in the holographic description (such as the mass and angular momentum of an AdS3 spacetime) to the periods of the quadratic differentials arising via the Bers embedding of T(1)xT(1). Our construction also yields a symplectic map from T*T(1) to T(1)xT(1) generalizing the well-known Mess map in the compact spatial surface setting.Comment: 41 pages, 2 figures, revised version accepted for publication in Commun.Math.Phy

Cosmological measurements, time and observables in (2+1)-dimensional gravity

We investigate the relation between measurements and the physical observables for vacuum spacetimes with compact spatial surfaces in (2+1)-gravity with vanishing cosmological constant. By considering an observer who emits lightrays that return to him at a later time, we obtain explicit expressions for several measurable quantities as functions on the physical phase space of the theory: the eigentime elapsed between the emission of a lightray and its return to the observer, the angles between the directions into which the light has to be emitted to return to the observer and the relative frequencies of the lightrays at their emission and return. This provides a framework in which conceptual questions about time, observables and measurements can be addressed. We analyse the properties of these measurements and their geometrical interpretation and show how they allow an observer to determine the values of the Wilson loop observables that parametrise the physical phase space of (2+1)-gravity. We discuss the role of time in the theory and demonstrate that the specification of an observer with respect to the spacetime's geometry amounts to a gauge fixing procedure yielding Dirac observables.Comment: 38 pages, 11 eps figures, typos corrected, references update

Notes on a paper of Mess

These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata.Comment: 26 page

Anti-de Sitter Quotients: When Are They Black Holes?

We point out that the BTZ black holes, and their relatives, can be defined in a cleaner way than they originally were. The covering space can be taken to be anti-de Sitter space, period, while scri splits up into components due to Misner singularities. Our definition permits us to choose between two conflicting claims concerning BTZ black holes in 3+1 dimensions.Comment: 16 pages, 4 figures; minor polish adde

A glimpse into Thurston's work

We present an overview of some significant results of Thurston and their impact on mathematics. The final version of this paper will appear as Chapter 1 of the book "In the tradition of Thurston: Geometry and topology", edited by K. Ohshika and A. Papadopoulos (Springer, 2020)

Anti-de Sitter geometry and Teichmüller theory

The aim of these notes is to provide an introduction to Anti-de Sitter geometry, with special emphasis on dimension three and on the relations with Teichm\"uller theory, whose study has been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art