Acyclic 2-dimensional complexes and Quillen's conjecture

Abstract

Let G be a finite group and Ap(G) be the poset of nontrivial elementary abelian p-subgroups of G. Quillen conjectured that Op(G) is nontrivial if Ap(G) is contractible. We prove that Op(G) 6= 1 for any group G admitting a G-invariant acyclic p-subgroup complex of dimension 2. In particular, it follows that Quillen’s conjecture holds for groups of p-rank 3. We also apply this result to establish Quillen’s conjecture for some particular groups not considered in the seminal work of Aschbacher–Smith.