Acyclic 2-dimensional complexes and Quillen’s conjecture

Piterman, Kevin Iván; Costa, Iván Sadofschi; Viruel, Antonio

doi:10.5565/publmat6512104

Cited by 4 publications

(6 citation statements)

References 9 publications

(19 reference statements)

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“…In [5] Aschbacher and Smith proved that a group satisfies (Q-QC) if > 5, and whenever has a unitary component U ( ) with ≡ −1 (mod ) and odd, then (QD) holds for all -extensions of U with ≤ and ∈ Z (see Definition 3.2). In a joint work with Sadofschi Costa and Viruel [14], we proved new cases of the conjecture not included in the previously mentioned results. We worked with the integer version of the conjecture (Z-QC) and proved that it holds if K S ( ) is homotopy-equivalent to a 2-dimensional and -invariant subcomplex.…”

Section: Introductionmentioning

confidence: 60%

“…has dimension 2 (rather than the dimension 3 of A 2 ( ) itself). This can be proved by using a similar argument to that of [14,Examples 4.10 and 4.11]. Finally, Corollary 4.7 applies since K( (A 2 ( ))) is a -invariant subcomplex of K(S 2 ( )) and homotopy-equivalent to K(A 2 ( )).…”

Section: Subcase 1a = a ( )mentioning

confidence: 67%

“…We close this section by using the rational version of Theorem 4.1 to extend some results of [14] on the integer conjecture (Z-QC) to the rational conjecture (Q-QC). We recall some of the main results of [14]. Proof.…”

Section: Now We Prove Theoremmentioning

confidence: 99%

“…We can also regard the poset N( ) as the 'neighbourhood' of A ( ), and N( ) − A ( ) as the 'boundary' of this neighbourhood. We give some consequences of this definition, which were used for computing the examples given in [14] (compare [18,19]). Next, we show that the elements outside N( ) attach to it via their centralisers in .…”

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confidence: 99%

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An approach to Quillen’s conjecture via centralisers of simple groups

Piterman

2021

Forum of Mathematics, Sigma

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For any given subgroup H of a finite group G, the Quillen poset ${\mathcal {A}}_p(G)$ of nontrivial elementary abelian p-subgroups is obtained from ${\mathcal {A}}_p(H)$ by attaching elements via their centralisers in H. We exploit this idea to study Quillen’s conjecture, which asserts that if ${\mathcal {A}}_p(G)$ is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the ${{\mathbb {Z}}}$ -acyclic version of the conjecture (obtained by replacing ‘contractible’ by ‘ ${{\mathbb {Z}}}$ -acyclic’). We also work with the ${\mathbb {Q}}$ -acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least $2$ . This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most $4$ .

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Section: Introductionmentioning

confidence: 60%

Section: Subcase 1a = a ( )mentioning

confidence: 67%

Section: Now We Prove Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

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An approach to Quillen’s conjecture via centralisers of simple groups

Piterman

2021

Forum of Mathematics, Sigma

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“…In the last few years, there have been further developments in the Quillen conjecture [15,16,17,18]. Recently, in [18], new tools for the study of the conjecture have been provided.…”

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confidence: 99%

Maximal subgroups of exceptional groups and Quillen's dimension

Piterman¹

2023

Preprint

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A. Given a finite group and a prime , let A ( ) be the poset of nontrivial elementary abelian -subgroups of . The group satisfies the Quillen dimension property at if A ( ) has non-zero homology in the maximal possible degree, which is the -rank of minus 1. For example, D. Quillen showed that solvable groups with trivial -core satisfy this property, and later, M. Aschbacher and S.D. Smith provided a list of all -extensions of simple groups that may fail this property if is odd. In particular, a group with this property satisfies Quillen's conjecture:has trivial -core and the poset A ( ) is not contractible.In this article, we focus on the prime = 2 and prove that the 2-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy the Quillen dimension property, with only finitely many exceptions. We achieve these conclusions by studying maximal subgroups and usually reducing the problem to the same question in small linear groups, where we establish this property via counting arguments. As a corollary, we reduce the list of possible components in a minimal counterexample to Quillen's conjecture at = 2.

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A geometric approach to Quillen’s conjecture

Ramos

Mazza

2021

Journal of Group Theory

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We introduce admissible collections for a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the Quillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of | G | \lvert G\rvert . Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex, Ann. of Math. (2) 137 (1993), 3, 473–529], our techniques are simpler.

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Acyclic 2-dimensional complexes and Quillen’s conjecture

Cited by 4 publications

References 9 publications

An approach to Quillen’s conjecture via centralisers of simple groups

An approach to Quillen’s conjecture via centralisers of simple groups

Maximal subgroups of exceptional groups and Quillen's dimension

A geometric approach to Quillen’s conjecture

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