Generalized Baumslag-Solitar groups are defined as fundamental groups of
graphs of groups with infinite cyclic vertex and edge groups. Forester proved
(in "On uniqueness of JSJ decompositions of finitely generated groups",
Comment. Math. Helv. 78 (2003) pp 740-751) that in most cases the defining
graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we
extend Forester's results to graphs of groups with vertex groups that can be
either infinite cyclic or quadratically hanging surface groups.Comment: 20 pages, 2 figures. Several corrections and improvements from
referee's report. Imprtant changes in Definition 5.1, and the proof of
Theorem 5.5 (previously 5.4). Lemma 5.4 was adde
