# Juan Alonso

### First Position

Postdoc at Uruguay University### Dissertation

*Graphs of Free Groups and their Measure Equivalence*

### Advisor

### Research Area

### Abstract

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 π_{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. This generalizes a result by Forester. 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. Then we characterize the quadratic Baumslag-Solitar groups that are measure equivalent to a free group, according to their defining graphs of groups.