A Phan-type theorem for Sp(2 n, q)

Abstract

In 1977 Kok-Wee Phan published a theorem (see [4]) on generation of the special unitary group SU(n+ 1, q2) by a system of its subgroups isomorphic to SU(3, q2). This theorem is similar in spirit to the famous Curtis–Tits theorem. In fact, both the Curtis–Tits theorem and Phan’s theorem were used as principal identification tools in the classification of finite simple groups. The proof of Phan’s theorem given in his 1977 paper is somewhat incomplete. This motivated Bennett and Shpectorov [1] to revise Phan’s paper and provide a new and complete proof of his theorem. They used an approach based on the concepts of diagram geometries and amalgams of groups. It turned out that Phan’s configuration arises as the amalgam of rank two parabolics in the flag-transitive action of SU(n + 1, q2) on the geometry of nondegenerate subspaces of the underlying unitary space. This point of view leads to a twofold interpretation of Phan’s theorem: its complete proof must include (1) a classification of related amalgams; and (2) a verification that—apart from some small exceptional cases—the above geometry is simply connected. These two parts are tied together by a lemma due to Tits, that implies that if a group G acts flag-transitively on a simply connected geometry then the corresponding amalgam of maximal parabolics provides a presentation for G, see Proposition 7.1. The Curtis–Tits theorem can also be restated in similar geometric terms. Let G be a Chevalley group. Then G acts on a spherical building B and also on the corresponding twin building B = (B+,B−, d∗). (Here B+ ∼= B ∼= B− and d∗ is a codistance between