## JSJ Decompositions

### Edited by org.aimpl.user:bestvina@math.utah.edu, org.aimpl.user:rscott@schubert.scu.edu, org.aimpl.user:Panagiotis.Papazoglou@maths.ox.ac.uk, and org.aimpl.user:morrison@aimath.org

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.

### Sections

Cite this as: *AimPL: JSJ Decompositions, available at http://aimpl.org/jsjdecomp.
*