Acyclic 2-dimensional complexes and Quillen's conjecture

2019/07/03 by Piterman, Kevin I., Costa, Iván Sadofschi, Viruel, Antonio
#20D05 #55M20 #55M35 #57M20 #57M60 #57S17 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1907.02141

Abstract

Let G be a finite group and A_p(G) be the poset of nontrivial elementary abelian p-subgroups of G. Quillen conjectured that O_p(G) is nontrivial if A_p(G) is contractible. We prove that O_p(G)≠ 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.

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