We review some infinite dimensional symmetries that appear indimensionally reduced classical general relativity. Reducing from d=4to d=2 gives a two dimensional nonlinear sigma model with an infinitedimensional affine Kac-Moody symmetry called the Geroch group.Further reducing to d=1 (along a null Kill vector) enhances the Gerochgroup to a mysterious "hyperbolic Kac-Moody" symmetry.
Further reading: P. Breitenlohner and D. Maison, "On the Geroch group," Ann. Inst.H. Poincare, 46, 215 (1987) H. Nicolai, "Two-dimensional gravities and supergravities asintegrable systems," in Schladming 1991, Proceedings, Recent aspectsof quantum fields, 231, and Hamburg DESY 91-038 H. Lu, M. Perry, and C. Pope, ArXiv:0711.0400 and ArXiv:0712.0615