Classification of Curtis-Tits and Phan amalgams with $3$-spherical diagram

TL;DR

This paper classifies all non-collapsing Curtis-Tits and Phan amalgams with 3-spherical diagrams over all fields, establishing uniqueness for spherical diagrams and linking amalgams to Kac-Moody groups.

Contribution

It provides a complete classification of certain amalgams with 3-spherical diagrams and characterizes those arising from Kac-Moody groups, including a finite presentation.

Findings

Amalgams with spherical diagrams are unique.

A necessary and sufficient condition for an amalgam to come from a Kac-Moody group.

A finite presentation for a large class of Kac-Moody type groups.

Abstract

We classify all non-collapsing Curtis-Tits and Phan amalgams with $3$-spherical diagram over all fields. In particular, we show that amalgams with spherical diagram are unique, a result required by the classification of finite simple groups. We give a simple condition on the amalgam which is necessary and sufficient for it to arise from a group of Kac-Moody type. This also yields a definition of a large class of groups of Kac-Moody type in terms of a finite presentation.