On some algebraic and analytic properties of the finitely generated simple left orderable groups $G_{\rho}$

In $2019$ Hyde and the second author constructed the first family of finitely generated, simple, left orderable groups. We prove that these groups are not finitely presentable, non-inner amenable, don't have Kazhdan's property $(T)$ (yet have property FA), and that their first $l^2$-Betti number vanishes. We also show that these groups are uniformly simple, providing examples of uniformly simple finitely generated left orderable groups. Finally, we also describe the structure of the groups $G_{\rho}$ where $\rho$ is a periodic labeling.