Acyclic $2$-dimensional complexes and Quillen's conjecture

Abstract

Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ 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.