On the Phan system of the Schur cover of SU(4, 32)

Abstract

This article is part of the program described in Bennett et al. (in: Ivanov et al. (ed.) Groups, combinatorics, and geometry, 2003) [3]. We study the Phan amalgams and their universal completions that occur for q = 3 in rank n = 3 for the diagram A 3 = D 3 , corresponding to SU(4, 3 2 ) ∼ = Spin − (6, 3). We show that the associated geometries admit universal 9-fold coverings, by showing that the universal completion of the Phan amalgam is the central extension of SU(4, 3 2 ) by its Schur multiplier. This information provides the last missing piece of information in the full classification of Phan amalgams and their universal completions for A n and D n .