Open problems and results in the group theoretic approach to quantum gravity via the BMS group and its generalizations

The Bondi-Metzner-Sachs group B is the common asymptotic group of all asymptotically flat (lorentzian) space-times, and is the best candidate for the universal symmetry group of General Relativity. However, in quantum gravity, complexified or euclidean versions of General Relativity are frequently considered. McCarthy has shown that there are forty-two generalizations of B for these versions of the theory and a variety of further ones, either real in any signature, or complex. A firm foundation for quantum gravity can be laid by following through the analogue of Wigner's programme for special relativity with B replacing the Poincare group P. Here the main results which have been obtained so far in this research programme are reported and the more important open problems are stated.