  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.
