This work concerns the Geometric Group Theory of an interesting class of groups that can be obtained as graphs of free groups. These groups are called Quadratic Baumslag-Solitar groups, and are defined by graphs of groups that have infinite cyclic edge groups, and whose vertex groups are either infinite cyclic, or surface groups [pi]1 (S ) that are "attached by their boundary", meaning that the edge groups of the adjacent edges correspond to the subgroups generated by the boundary classes of S . More generally, we may also take S to be a 2-orbifold. The first part of the thesis studies JSJ decompositions for groups. We prove that, in most cases, the defining graph of groups of a Quadratic Baumslag-Solitar group is a JSJ decomposition in the sense of Rips and Sela [36]. This generalizes a result by Forester [11]. The second part studies measure equivalence between groups. It involves the concept of measure free factors of a group, which is a generalization of that of free factors, in a measure theoretic context. We find new families of cyclic measure free factors of free groups and some virtually free groups, following a question by D. Gaboriau [16]. Then we characterize the Quadratic Baumslag-Solitar groups that are measure equivalent to a free group, according to their defining graphs of groups.