We will show how minimal dynamical systems and etale groupoids can be used to construct finitely generated simple groups with prescribed properties. For example, one can show that there are uncountably many different growth types (in particular quasi-isometry classes) among finitely generated simple groups, or embed the Grigorchuk group into a simple torsion group of intermediate growth. Other properties like torsion and amenability will be also discussed.[-]

We show that several new classes of groups are measure strongly treeable, i.e., all of their free measure-class-preserving actions are treeable. This includes all finitely generated groups admitting planar Cayley graphs, all finitely generated elementarily free groups, and more generally all groups arising as the fundamental group of an 'IFL tower' over these groups. Our techniques also lead to new measure strong free factors of groups, i.e., group elements which generate a primitive subrelation in every free measure-class-preserving action. This is based on joint work with Clinton Conley, Damien Gaboriau, and Andrew Marks.[-]