RGD-systems of type $(4,4,4)$ over ${\mathbb{F}}_2$ and tree products

In this paper we prove that the group $U_{+}$ of an RGD-system of type $(4,4,4)$ over ${\mathbb{F}}_2$ contains a certain tree product as a subgroup. The proof relies on a careful analysis of the action on the associated twin building. This result is part of a larger project and we will use it as an induction start to construct uncountably many RGD-systems of type $(4,4,4)$ over ${\mathbb{F}}_2$.