In a joint project with Ilaria Castellano and José Pedro Quintanilha, we show how one can, from a nice action of a group on a product of heighted trees, deduce the full Sigma theory of the group. This talk is about our quest for interesting examples of such actions.

I will introduce a class of groups that generalises classical Baumslag-Solitar groups. We will see that these groups arise as iterated HNN extensions, and have a subgroup structure akin to parabolic subgroups in Artin Groups. Finally, we will see how these groups act on products of Bass-Serre trees and give a nice description of these products.