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.Comment: Talk given at Geometry and Physics in Cracow, September 2010; 22
pages, 2 figure
