A Curtis-Tits-Phan theorem for the twin-building of type \(A_{n-1}\) (Q1014598)

The aim of the paper under review is to characterize the groups \(G=\mathrm{SU}_n(k[t,t^{-1}],\beta)\) as universal completions of certain amalgams. Here \(n\geq 4\), \(k\) is an infinite field or a finite field of order \(q^2\) with \(q\geq 4\), \(\beta\) is a \(\sigma\)-hermitian form on an \(n\)-dimensional vector space over the field of rational functions \(k(t)\) where \(\sigma\) is an involutory automorphism of \(k(t)\) that preserves \(k\) and such that its restriction to \(k\) is not the identity, such that the associated norm function from \(k\) onto the fixed field of \(\sigma\) is surjective and such that \(t^\sigma=t^{-1}\). Equivalently, the group \(G\) has a weak Phan system of type \(\tilde A_{n-1}\), and any group admitting a weak Phan system isomorphic (as an amalgam) with the one in \(G\) is a quotient of \(G\). In this form it extends a similar result by \textit{C. D. Bennett} and \textit{S. Shpectorov} [J. Group Theory 7, 287--310 (2004; Zbl 1055.20022)] for the special unitary groups over finite fields. The authors use a similar approach as Bennett and Shpectorov and proceed geometrically by finding a suitable simply connected geometry on which \(G\) acts flag-transitively. To obtain the required geometry they start with the lattice-chain model for the affine twin building \(\Delta\) of type \(\tilde A_{n-1}\) over \(k(t)\). The involution \(\sigma\) induces a geometric flip \(\varphi\) for \(\Delta\) and the residues of the associated flip-flop geometry \(\Gamma_\varphi\) are described in terms of \(\beta\). Using a criterion for simple connectedness of the flip-flop geometry associated to a flip of a twin-building by \textit{A. Devillers} and \textit{B. Mühlherr} [Forum Math. 19, 955--970 (2007; Zbl 1188.51003)], it is shown that under the assumptions made, \(\Gamma_\varphi\) and all residues of rank at least 3 (with a few exceptions) are simply connected. Since \(G\) acts flag-transitively on \(\Gamma_\varphi\), the result for \(G\) follows from Tits' Lemma.