A Phan-type Theorem for Orthogonal Groups

Phan’s theorem and the Curtis-Tits’ theorem are useful tools in the original proof of the Classification of Finite Simple Groups and the ongoing Gorenstein-Lyons-Solomon revision. Bennett, Gramlich, Hoffman and Shpectorov proved in a series of papers that Phan’s theorem and the Curtis-Tits’ theorem were results with very geometric proofs. They created a technique to prove these results which was generalized to produce what they called Curtis-Phan-Tits Theory. The present paper applies this technique to the orthogonal groups. A geometry is created on which a particular orthogonal group acts flag-transitively. The geometry is shown to be both connected and then simply connected when the dimension of the orthgonal group is at least five (except when the field is order three). After these facts are established Tits’ lemma is used to conclude that the orthogonal group is the universal completion of an interesting amalgam of subgroups that is associated with the geometry. This type of result is useful in the context of identifying a group when there is knowledge of the subgroup structure