Complexity volumes of splittable groups
  Using graph of groups decompositions of finitely generated groups, we define Euler characteristic type invariants which are non-zero in many interesting classes of finitely presented, hyperbolic, limit and CSA groups, including elementarily free groups and one-ended torsion-free hyperbolic groups whose JSJ decomposition contains a maximal hanging Fuchsian vertex group. 
Mihalis Sykiotis
math.GR