Consequences of a quasi-polynomial time algorithm for the graph isomorphism problem The Graph Isomorphism problem (GI) is arguably 
the best known candidate for an NP-intermediate problem. 
The best known algorithm is sub-exponential algorithm 
with run-time $2^{O(\sqrt{n \log n})}$. 
It is known that GI is not $\mathsf{NP}$-complete unless 
the polynomial hierarchy collapses. 


  What would be the complexity theoretic consequences of a quasi-polynomial time algorithm for the Graph Isomorphism problem?
  Would a quasi-polynomial time algorithm for GI refute any famous conjectures in complexity theory?




Other similar problems like 
Minimum Dominating Set in Tournaments problem, 
Group Isomorphism problem, and 
Tournament Isomorphism problem 
have quasi-polynomial time (QP) algorithms. 
The later two problems are polynomial-time reducible to GI.


  Can we efficiently reduce Minimum Dominating Set in Tournaments problem to GI?
  Is there any conjecture ruling out GI being hard for QP?


Update (2015-12-14): Babai has posted a preliminary draft paper on arXiv for his quasipolynomial-time algorithm for GI.

Update (2017-01-04): Babai retracted the claim that the algorithm is in quasipolynomial time, according to the new analysis the algorithm is in subexponential time $\exp \exp(\tilde{O}(\sqrt{\lg n}))$ which is inside $2^{n^{o(1)}}$.

Update (2017-01-09): Babai reinstated the quasipolynomial time claim, replacing the offending procedure with a more efficient one.
As far as I can tell, if you ask simply about the consequences of the mere fact (as a black box) that GI is in QP, I think the answer is very little. The one thing I can think of, which is not a theorem but a consequence for research directions, is to Group Isomorphism. Since GroupIso reduces to GI and we don't even know if GroupIso is in P, putting GroupIso into P can be seen as an important obstacle to putting GI into P (if you think the latter might be the case). 

However, since the trivial algorithm for GroupIso is $n^{\log n + O(1)}$, back when the complexity of GI was up at $2^{\tilde O(\sqrt{n})}$, we had a long way to go in improving the complexity of GI before GroupIso became an immediately relevant obstacle to putting GI into P. But if GI is in QP, then GroupIso becomes a much more relevant obstacle to further improvements in GI. (Of course, the exponent of the exponent in the quasi-polynomial is still a potentially relevant gap, but the gap becomes a lot smaller when GI is in QP.)