A Classification of Curtis-Tits Amalgams

Abstract

A celebrated theorem of Curtis and Tits on groups with finite BN-pair shows that these groups are determined by the local structure arising from their fundamental subgroups of ranks $1$ and $2$. This result was later extended to Kac-Moody groups by P. Abramenko and B. Mühlherr and Caprace. Their theorem states that a Kac-Moody group $G$ is the universal completion of an amalgam of rank two (Levi) subgroups, as they are arranged inside $G$ itself. Taking this result as a starting point, we define a Curtis-Tits structure over a given diagram to be an amalgam of groups such that the sub-amalgam corresponding to a two-vertex sub-diagram is the Curtis-Tits amalgam of some rank-$2$ group of Lie type. There is no a priori reference to an ambient group, nor to the existence of an associated (twin-) building. Indeed, there is no a priori guarantee that the amalgam will not collapse. We then classify these amalgams up to isomorphism. In the present paper we consider triangle-free simply-laced diagrams. Instead of using Goldschmidt’s lemma, we introduce a new approach by applying Bass and Serre’s theory of graphs of groups, not to the amalgams themselves but to a graph of groups consisting of certain automorphism groups. The classification reveals a natural division into two main types: “orientable” and “non-orientable” Curtis-Tits structures. Our classification of orientable Curtis-Tits structures naturally fits with the classification of all locally split Kac-Moody groups over fields with at least four elements using Moufang foundations. In particular, our classification yields a simple criterion for recognizing when Curtis-Tits structures give rise to Kac-Moody groups. The class of non-orientable Curtis-Tits structures is in some sense much larger. Many of these amalgams turn out to have non-trivial interesting completions inviting further study.

Subject Classification: [2010] Primary 20G35; Secondary 51E24, 20E42The authors wish to thank the Banff International Research Station, Banff, Canada for a very productive visit as part of their Research In Teams program (08rit130).