A classification of Curtis-Tits amalgams

TL;DR

This paper classifies Curtis-Tits amalgams over triangle-free simply-laced diagrams, revealing two main types and providing criteria for their relation to Kac-Moody groups, using Bass-Serre theory instead of Goldschmidt's lemma.

Contribution

It introduces a new classification approach for Curtis-Tits amalgams using Bass-Serre theory, distinguishing orientable and non-orientable types, and connects to Kac-Moody group structures.

Findings

Classification into orientable and non-orientable types.

A criterion for when amalgams give rise to Kac-Moody groups.

Many non-orientable amalgams have interesting non-trivial completions.

Abstract

A celebrated theorem of Curtis and Tits on groups with finite BN-pair shows that roughly speaking these groups are determined by their local structure. This result was later extended to Kac-Moody groups by P.~Abramenko and B.~M\"uhlherr. 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. 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 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.