The sphericity of the Phan geometries of type Bn and Cn and the Phan-type theorem of type F4

TL;DR

This paper proves the sphericity of certain Phan geometries, determines finiteness lengths of related groups, and establishes the first proof of the Phan-type theorem of type F4, extending classification results.

Contribution

It adapts methods to prove sphericity of Phan geometries of types Bn and Cn, and provides the first proof of the Phan-type theorem of type F4, extending classification of finite simple groups.

Findings

Proved sphericity of Phan geometries of types Bn and Cn.

Determined finiteness length of certain hyperbolic Kac--Moody groups.

First published proof of the Phan-type theorem of type F4.

Abstract

We adapt and refine methods developed by Abramenko and Devillers--K\"ohl--M\"uhlherr in order to establish the sphericity of the Phan geometries of type B_n and C_n, and their generalizations. As an application we determine the finiteness length of the unitary form of certain hyperbolic Kac--Moody groups. We also reproduce the finiteness length of the unitary form of the groups Sp_{2n}(GF(q^2)[t,t^{-1}]). Another application is the first published proof of the Phan-type theorem of type F_4. Within the revision of the classification of the finite simple groups this concludes the revision of Phan's theorems and their extension to the non-simply laced diagrams. We also reproduce the Phan-type theorems of types B_n and C_n.