Presentation of hyperbolic Kac–Moody groups over rings

Tits has defined Kac–Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley–Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac–Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis–Tits presentation of the finite groups of Lie type. When the ground ring is finitely generated, we derive the finite presentability of the Steinberg group, and similarly for the Kac–Moody group when the ground ring is a Dedekind domain of arithmetic type. These finite-presentation results need slightly stronger hypotheses when the rank is smallest possible, namely 4. The presentations simplify considerably when the ground ring is ZZ, a case of special interest because of the conjectured role of the Kac–Moody group E10(Z)E10(Z) in superstring theory.