Finite presentability of Kac-Moody groups over finite fields

Abstract

Let $\mathcal{G}$ be a Kac-Moody group functor in the sense of Tits, with associated Coxeter system $(W,S)$. For any field $F$, the group $\mathcal{G}(F)$ is finitely generated iff $F$ is finite. We are interested in the question when $G = \mathcal{G}(\mathbb{F}_q)$ is finitely presented. If $(W,S)$ is 2-spherical, it is well known that this is "almost always" the case. It is conjectured that $G$ is never finitely presented if $(W,S)$ is not 2-spherical (which means that there exist $s,t \in S$ with $|st| = \infty$), which so far (to the best of our knowledge) has only been proved for the type $\tilde{A_1}$, and maybe also, though we don't know a reference for this, in the case where $|st| = \infty$ for all $s \neq t \in S$. In this paper, we show that $G$ is not finitely presented for a significantly larger class of Coxeter systems which are not 2-spherical, giving much stronger evidence that the conjecture is true in general. Important tools of the proof are the twin BN-pair and the corresponding twin building associated to $G = \mathcal{G}(\mathbb{F}_q)$.