Quasi-isometry Invariants from Decorated Trees of Cylinders of Two-Ended   JSJ Decompositions
  We construct quasi-isometry invariants of a one-ended finitely presented group by considering the tree of cylinders of a two-ended JSJ decomposition of the group. When the group satisfies additional quasi-isometric rigidity hypotheses we construct finer invariants by also considering relative amounts of stretching across edges of the tree of cylinders. 
Christopher H. Cashen
math.GR