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.Comment: Major revisions following the suggestions of the referee. To appear
in Journal of Algebr
