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.
Funding Statement
This work was partially done at the University of Málaga, during a research stay of the first two authors, supported by project MTM2016-78647-P. The first author was supported by a CONICET doctoral fellowship and grants CONI- CET PIP 112201701 00357CO and UBACyT 20020160100081BA. The second author was supported by a CONICET postdoctoral fellowship and grants ANPCyT PICT-2017-2806, CONICET PIP 11220170100357CO, and UBACyT 20020160100081BA. The third author was partially supported by Ministerio de Economía y Competitivi- dad (Spain), grant MTM2016-78647-P (AEI/FEDER, UE, support included).
Citation
Kevin Iván Piterman. Iván Sadofschi Costa. Antonio Viruel. "Acyclic 2-dimensional complexes and Quillen’s conjecture." Publ. Mat. 65 (1) 129 - 140, 2021. https://doi.org/10.5565/PUBLMAT6512104
Information
