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The last decades there has been a lot of research going on in the the area of three dimensional gravity. The motivation, I understand, is threefold:

  1. Whereas gravity is not perturbatively renormalizable in four spacetime dimensions, in three dimensions it is. To make it even more interesting it has black hole solutions and it is exactly solvable. This opens the way to to study quantum black holes. This make three dimensional gravity a very interesting system on itself.

  2. Through the AdS/CFT correspondence there is a connection between conformal field theories (CFT) in two dimensions and gravity in three dimensions. CFT's are important in condensed matter physics and one can use 3D gravity to learn more about them.

  3. Gravity in three dimensions is simpler to deal with then gravity in four dimensions. Therefor it can be used as a toy model for gravity in four dimensions.

I am wondering what are the most important insights that 3d gravity brought in these respects? In particular I am interested in point three: did 3d gravity provide any new view on 4d gravity so far?

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This is a very good question. Everything that you mentioned is indeed true, with some controversy over the exactly solvable part for 3d gravity. Like you already mentioned, gravity in three dimensions is very different from higher dimensional gravity because of the fact that the graviton propagator is vanishing. In the 1980's following the work of Townsend, Achucarro (sorry if you need special access to get to this paper) and the work of Witten, it was accepted that 3d gravity is written in terms of a Chern - SImons theory with a yang-Mills type connection which could allow you to correctly determine the structure and features of the theory by studying the coset Wess-Zumino-Witten construction in a conformal field theory. There was also the discovery of the BTZ black hole by by Banados, Bunster and Zanelli which was very interesting and unexpected development. What this meant for the theory of quantum gravity in 2+1 dimensions was that the BTZ black hole entropy would have to be correctly reproduced by the quantum theory. Following the work of Brown and Henneaux, it was shown that the asymptotic symmetry group for the $AdS_3$ spacetime is the $Vir_c$ i.e. it is generated by the Virasoro group and therefore, by looking at global diffeomorphisms in $AdS_3$ which preserve the asymptotic structure of spacetime, you could define the Virasoro algebra and thereby calculate the central charges and then use the Cardy technique to determine the entropy. So all this gave a lot of confidence that gravity could be exactly solvable in 3 dimensions and the quantum version is trivial.

However, following some of the more recent work of Witten (which Witten claims that he is unsure if correct), he claims that the CFT dual is not so trivial and can in principle have a symmetry group which is the Meuermann, Lepowsky and Frenkel monster symmetry group for which the whole problem becomes much more complicated.Furthermore, new problems in the study of 2+1 d gravity are also that there does not seem to be a quantum theory of gravity which correctly reproduces the BTZ black hole entropy and also reduced to the Einstein Hilbert gravity in the classical limit.

But the place where your question is not valid is when you ask if whether 3d gravity can tell us anything about 4d gravity. The answer to this is no, at least not in any way that anyone has thought about so far. This is again because of the seemingly more complex and intricate structure of AdS space in 2+1 dimensions and the possibly large symmetry group for the CFT. Furthermore, even from the classical limit, there really isn't much that 3d gravity can tell you about 4d gravity because of the propagator structures in both the theories.

Hope what you wanted is contained in this answer. If not, I'll be happy to help!

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