Finite Generation of RGD Systems with Exceptional Links

Let $(G,(U_\alpha)_{\alpha\in \Phi},T)$ be an RGD system. The most prominent examples are Kac-Moody groups, which are infinite dimensional analogs of semisimple Lie groups. These groups have an associated twin building $\Delta$ on which the group $G$ will act strongly transitively. We say that the building $\Delta$ satisfies condition (co) if the collection of chambers opposite any chamber is gallery connected. It is known that if $\Delta$ satisfies (co), the subgroup $U_+$ of $G$ is generated by some finite set of fundamental root groups, and thus is finitely generated if these root groups are finitely generated.

We will help close the gap in the literature relying on condition (co) by proving when the subgroup $U_+$ of an RGD system associated to rank-3 buildings is finitely generated. Most of the time, the group $U_+$ will not be finitely generated, and we will give sufficient conditions to guarantee the infinite generation of $U_+.$ We will then modify this aproach to see that another group not covered by the conditions is also not finitely generated. Our main strategy will be to produce a large family of surjective homomorphisms from $U_+$ which send relatively few $U_\alpha$ to non-identity elements, implying that some of these $U_\alpha$ must be in any generating set.

Finally, we will show that there are two cases where $U_+$ remains finitely despite $\Delta$ not satisfying (co). We will use an aproach which relies on defining a distance between root groups, and showing that most root groups can be expressed in terms of those closer to the fundamental chamber. This approach can also give another proof of the finite generation of $U_+$ with condition (co).