Simple connectivity of the Quillen complex of \(\text{GL}_ n(q)\) (Q1904078)

The Quillen complex \({\mathcal A}_p(G)\) of a finite group \(G\) at a prime \(p\) dividing the order of \(G\) is the order complex of the poset of nontrivial elementary Abelian \(p\)-subgroups of \(G\), ordered by inclusion. Known properties of \({\mathcal A}_p(G)\) and the result of Gorenstein and Lyons are indicating that the pairs \(G\), \(p\) for which \({\mathcal A}_p(G)\) is disconnected are completely classified. According to Aschbacher, the question about the simple connectivity of \({\mathcal A}_p(G)\) has, to a large extent, been reduced to a minimal class of groups, which includes the classical groups of Lie type. It is proved, that \({\mathcal A}_p(G)\) for \(G=\text{GL}(n,q)\) is simply connected whenever \(m_p(G)(=\max\{m_p(A)\mid A\in{\mathcal A}_p(G)\})\geq3\). This was known by results of Quillen, Solomon and Tits for the cases when \(p\) is the characteristic prime, and when \(p\mid q-1\). If \(p\) is not a divisor for \(q-1\), then \({\mathcal A}_p(G)\) coincides with \({\mathcal A}_p(\text{SL}(n,q))\) and is isomorphic to \({\mathcal A}_p(\text{PSL}(n,q))\).