  We perform a canonical quantization of pure gravity on AdS3 using as a
technical tool its equivalence at the classical level with a Chern-Simons
theory with gauge group SL(2,R)xSL(2,R). We first quantize the theory
canonically on an asymptotically AdS space --which is topologically the real
line times a Riemann surface with one connected boundary. Using the "constrain
first" approach we reduce canonical quantization to quantization of orbits of
the Virasoro group and Kaehler quantization of Teichmuller space. After
explicitly computing the Kaehler form for the torus with one boundary component
and after extending that result to higher genus, we recover known results, such
as that wave functions of SL(2,R) Chern-Simons theory are conformal blocks. We
find new restrictions on the Hilbert space of pure gravity by imposing
invariance under large diffeomorphisms and normalizability of the wave
function. The Hilbert space of pure gravity is shown to be the target space of
Conformal Field Theories with continuous spectrum and a lower bound on operator
dimensions. A projection defined by topology changing amplitudes in Euclidean
gravity is proposed. It defines an invariant subspace that allows for a dual
interpretation in terms of a Liouville CFT. Problems and features of the CFT
dual are assessed and a new definition of the Hilbert space, exempt from those
problems, is proposed in the case of highly-curved AdS3.
