What is the definition of an Asymptotic Symmetry Group (ASG) of a spacetime? What is the definition of an Asymptotic Symmetry Group (ASG) of a spacetime? 
 A: Asymptotic Symmetry Groups (ASG) means symmetry groups at asymptotic infinity. But then you have to define asymptotic infinity in such a way that you can calculate. In General Relativity (GR) they are important because by having some timelike symmetry at infinity you can the conclude that a mass or energy conservation law can be defined for the mass/energy inside the volume surrounded by asymptotic infinity. 
There have been two ways in which that asymptotic infinity has been defined mathematically so it makes sense physically and you can calculate. One is spatial infinity, and the other lightlike infinity. 
If spacetime is flat at spacetime infinity the ASG is the Poincare group, and the conserved mass or energy is the ADM mass or energy. A flat space like infinity gives you all the Minkowski symmetries out there, in GR called Killing vectors, and they can be defined rigorously. In fact for the ADM mass you only need a timelike Killing vector at space like infinity. 
The other mass (or energy, I'll just say mass from now on meaning the same) is due to the BMS group which is a symmetry for flat spacetime at lightlike infinity. This is also called a conformal symmetry. BMS was derived by Bondi, Metzner, Van derBurg and Sachs (somehow BMS stuck), and the mass is called either the BMS mass or the Bondi mass. For only mass conservation again you only need the timelike Killing vector. BMS includes the Poincare groups and what are called supertranslations (and I don't recall, maybe also superrotations and  superboosts but not sure). 
In either case the groups at infinity (the ASGs) are defined in terms of asymptotic Killing vector fields. M 
See the discussion of the different masses at https://en.wikipedia.org/wiki/Mass_in_general_relativity
The BMS group and conformal infinity has gotten a lot more interest in the last few years. First because of the interest in CFT and the AdS/CFT correspondence, and since January or February of this year because of the work of Hawking, Perry and Strominger on the possibility and some results that those symmetries at lightlike infinity involve the creation of those conserved quantities at the horizons of Black Holes, and they hypothesize that the conserved quantities is the information that went into the Black Hole. They say that those are Soft Hair that the horizons have. Their paper is preliminary (not full results), but has some preliminary and interesting first results.  See their paper arXiv in January 2016, and published in Phys Rev Letters in June 2016. See the latter at http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.231301. ArXiv is at https://arxiv.org/abs/1601
See about conformal infinity and BMS in GR at http://relativity.livingreviews.org/open?pubNo=lrr-2004-1&page=articlesu2.html. 
The Wald book coverage of BMS (way before the Hawking latest paper) is at (yes, you have to really buy the book, hard to read for free) https://books.google.com/books?id=9S-hzg6-moYC&pg=PA284&lpg=PA284&dq=asymptotic+symmetry+group+of+spacetime&source=bl&ots=FHRMsJiG1b&sig=9LjXx5jn6IDaiZk2pyTFCiLVOZk&hl=en&sa=X&ved=0ahUKEwjmqMPDnJXPAhXBpYMKHUvJAKoQ6AEIPjAJ#v=onepage&q=asymptotic%20symmetry%20group%20of%20spacetime&f=false
THere is also a paper on the BMS group for deSitter spacetimes. 
