A rank 3 tangent complex of \(\text{PSp}_4(q)\), \(q\) odd (Q2761649)

The authors construct an incidence geometry \(\mathcal G\) from the group \(G=\text{PSL}_4(q)\) where \(q\) is an odd prime power. The points of \(\mathcal G\) are the root subgroups of \(G\): long, short and virtual. The lines of \(\mathcal G\) come in various forms one of which can be viewed as the `tangent bundle' of the \((q,q)\)-generalised quadrangle for \(G\) when this quadrangle is represented as a system of conics whose points are the long subgroups of \(G\). The flag complex of \(\mathcal G\) is residually connected and hence a rank 3 chamber complex. The correspondence between \(\mathcal G\) and the action of root elements in \(G\) on a symplectic module is then described explicitly.NEWLINENEWLINENEWLINEIn the last section the authors determine the subgroup generated by a pair of short-root groups not contained in a common parabolic subgroup. Using incidence relations in \(\mathcal G\) they directly show that such a pair generates the subgroup of index 2 in the centraliser of an involution of class either 2.A or 2.C. These centralisers are the plus-point and minus-point stabilisers, respectively, in the natural representation of \(G\) and both are maximal subgroups of \(G\). Maximal subgroups of \(\text{PSp}_{2n}(F)\) that contain short-root subgroups have been investigated by \textit{S.-Z. Li} [Acta Math. Sin., New Ser. 3, 82-91 (1987; Zbl 0642.20038)]. If \(q\neq 3\), the authors furthermore show that such a subgroup generated by a pair of short-root groups not contained in a common parabolic subgroup is isomorphic to either \(L_2(q^2)\) or \(2.L_2(q)\times L_2(q)\).