JSJ Decompositions

Splitting theory of groups parallels decomposition theorems in 3-manifold theory. Kneser’s theorem has its analog in Grushko’s theorem, Haken finiteness and hierarchies gave rise to accessibility and strong accessibility in group theory while JSJ decompositions of 3-manifolds were shown recently to extend to group theory too. One should stress that these "generalizations" from 3-manifolds to groups were by no means routine and a lot of questions remain open today. The structure theory that one obtains for groups is often much more general that the 3-manifold theory and it had many striking applications (isomorphism problem for hyperbolic groups, Tarski problems etc). We refer to [MR1469317] and [MR1664694] for the basic definitions.