J an 2 00 9 THE SPHERICITY OF THE PHAN GEOMETRIES OF TYPE B n AND C n AND THE PHAN - TYPE THEOREM OF TYPE F 4

We adapt and refine the methods developed in [Abr96] and [DGM] in order to establish the sphericity of the Phan geometries defined in [BGHS07] (type Bn) and [GHS03] (type Cn), and their generalizations. As applications of this sphericity we determine the topological finiteness length of the unitary form of the group Sp 2n (F q2 [t, t]) (Theorem 7.1) and give the first published proof of the Phan-type theorem of type F4 (Theorem 7.11). Apart from that we reproduce the topological finiteness length of the group Sp 2n (F q2 [t]) and the Phan-type theorems of types Bn and Cn. In the theory of arithmetic groups our result on the topological finiteness length of the unitary form of Sp 2n (F q2 [t, t]) is another example supporting the rank conjecture, see [Beh98, p. 80]. Within the revision of the classification of the finite simple groups this publication of the Phan-type theorem of type F4 concludes the revision of Phan’s theorems [Pha77a], [Pha77b] and their extension to the non-simply laced diagrams; cf. [AB08, Section 14.2] on page 656 and [GLS05] on page 333.