Galilean quantum gravity in 2+1 dimensions

In this thesis, we study the Galilean limit of gravity in 2+1 dimensions and give the necessary ingredients for its quantisation. We study two groups that play fundamental role in this thesis, the two-fold central extension of the Galilei and Newton-Hooke groups in 2+1 dimensions and their corresponding Lie algebras. We construct what we call \Galilean gravity in 2+1 dimensions" as the Chern-Simons theory of the Galilei group and generalise this construction to include a cosmological constant which, in the present setting corresponds to the Chern-Simons theory of the Newton-Hooke group. Finally, we apply the combinatorial quantisation program in detail to the Galilei group: we give the irreducible, unitary representation of the relevant quantum double and fully explore Galilean quantum gravity in this setting. We highlight the associated structures for the Newton-Hooke group and provide an outline for a similar quantisation. In doing so, we provide the link between Newton-Hooke gravity, and a deformation of an extension of the Heisenberg algebra that is well-studied