A Curtis–Tits–Phan theorem for the twin-building of type A˜n−1

AbstractThe Curtis–Tits–Phan theory as laid out originally by Bennett and Shpectorov describes a way to employ Tits' lemma to obtain presentations of groups related to buildings as the universal completion of an amalgam of low-rank groups. It is formulated in terms of twin-buildings, but all concrete results so far were concerned with spherical buildings only. We describe an explicit flip–flop geometry for the twin-building of type A˜n−1 associated to k[t,t−1] on which a unitary group SUn(k[t,t−1],β), related to a certain non-degenerate hermitian form β, acts flag-transitively and obtain a presentation for this group in terms of a rank-2 amalgam consisting of unitary groups. This is the most natural generalization of the original result by Phan for the unitary groups