While the ultraviolet behavior of quantum field theories was tamed by the renormalization program, the infrared structure was for a long time considered to be trivial and uninteresting. Nevertheless, it turns out this is not the case. One has very interesting behavior at the infrared limit.
Firstly one has the so-called Bondi—Metzner—Sachs (BMS) group, which is the symmetry group at null infinity of asymptotically flat spacetimes. In principle, one would expect this group to be simply the Poincaré group (and hence the symmetries at infinity are just the symmetries of Minkowski spacetime), but actually one gets an infinite-dimensional Lie group comprised of the Poincaré transformations and the so-called supertranslations, which are direction-dependent translations. This means general relativity does not reduce to special relativity at very large distances, but rather to something much more complicated. The existence of this infinite-dimensional symmetry group in the boundary of the spacetime can be exploited to obtain more information about what happens in the bulk.
Secondly one has Weinberg’s soft graviton theorem. A soft particle is a particle with very small energy. Weinberg’s soft graviton theorem relates a scattering amplitude to the amplitude for the same scattering with the addition of a number of soft gravitons. This turns out to be a consequence of the BMS symmetries at null infinity, and it is fundamental to fully understand scattering in quantum field theories involving gravitons or gauge fields.
Thirdly, one has the memory effect. This is the prediction that two nearby inertial detectors close to infinity will be permanently displaced by the passage of a gravitational wave. The computations involved in the prediction of this effect can be matched to the soft graviton theorem, and the effect can be understood as a physical realization of a BMS translation. This effect is expected to be measured in future gravitational wave detectors.
The connections between these three topics allow one to study one of them to gather information about the other two, leading into further insights on the behavior of quantum gravity at very low energies.
The following references discuss the infrared structure of gravity and gauge theories.