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

## TL;DR

This paper proves Quillen's conjecture for groups with a specific acyclic 2-dimensional complex structure, confirming the conjecture for p-rank 3 groups and some additional cases, advancing understanding of group subgroup posets.

## Contribution

It demonstrates that groups with a G-invariant acyclic 2-dimensional p-subgroup complex satisfy Quillen's conjecture, especially confirming it for p-rank 3 groups and certain other groups.

## Key findings

- Proves $O_p(G)
eq 1$ for groups with a G-invariant acyclic 2-dimensional p-subgroup complex.
- Confirms Quillen's conjecture for groups of p-rank 3.
- Establishes Quillen's conjecture for specific groups beyond previous results.

## 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.