Quantum gravity and non-commutative spacetimes in three dimensions: a unified approach

These notes summarise a talk surveying the combinatorial or Hamiltonian quantisation of three dimensional gravity in the Chern-Simons formulation, with an emphasis on the role of quantum groups and on the way the various physical constants (c,G,\Lambda,\hbar) enter as deformation parameters. The classical situation is summarised, where solutions can be characterised in terms of model spacetimes (which depend on c and \Lambda), together with global identifications via elements of the corresponding isometry groups. The quantum theory may be viewed as a deformation of this picture, with quantum groups replacing the local isometry groups, and non-commutative spacetimes replacing the classical model spacetimes. This point of view is explained, and open issues are sketched.