A geometric approach to Quillen's conjecture (Q2068341)

Given a finite group \(G\) and a prime number \(p\) dividing the order of \(G,\) let \(\mathcal A_p(G)\) be the poset of all non-trivial elementary abelian \(p\)-subgroups of \(G\) ordered by inclusion, and let \(|\mathcal A_p(G)|\) be its realization as a topological space. In his seminal paper, \textit{D. Quillen} [Adv. Math. 28, 101--128 (1978; Zbl 0388.55007)] conjectured that if \(|\mathcal A_p(G)|\) is contractible, then \(G\) contains a non-trivial normal \(p\)-subgroup. \textit{M. Aschbacher} and \textit{S. D. Smith} [Ann. Math. (2) 137, No. 3, 473--529 (1993; Zbl 0782.20039)], tackling Quillen's conjecture, introduced for this purpose a stronger version, called the Quillen dimension at \(p\) property, involving reduced homology with rational coefficients. The authors introduce admissible collections for a finite group \(G\) and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the Quillen dimension at \(p\) property, when \(p\) is an odd prime.