The Pure Symmetric Automorphisms of a Free Group Form a Duality Group

Brady, Noel; McCammond, Jon; Meier, John; Miller, Andy

doi:10.1006/jabr.2001.8944

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Introduction4

By [Mccool 1986] G Is Finitely Presented With Relations1

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Cited by 15 publications

(16 citation statements)

References 12 publications

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“…By [Brady et al 2001], G is a duality group of dimension n −1. By the main result of [Orlandi-Korner 2000], the class [χ ] is in 1 (G) c if and only if exactly one of the following holds:…”

Section: By [Mccool 1986] G Is Finitely Presented With Relationsmentioning

confidence: 99%

Sigma theory and twisted conjugacy classes

Gonçalves¹,

Kochloukova²

2010

Pacific J. Math.

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Section: By [Mccool 1986] G Is Finitely Presented With Relationsmentioning

confidence: 99%

Sigma theory and twisted conjugacy classes

Gonçalves¹,

Kochloukova²

2010

Pacific J. Math.

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“…See [Dam16] for a discussion of the many guises of these groups. Some topological properties known for PΣAut n include that is has cohomological dimension n − 1 [Col89], it is a duality group [BMMM01] and its cohomology ring has been computed [JMM06].…”

Section: Introductionmentioning

confidence: 99%

Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties

Zaremsky¹

2018

Michigan Math. J.

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The BNSR-invariants of a group G are a sequence Σ 1 (G) ⊇ Σ 2 (G) ⊇ · · · of geometric invariants that reveal important information about finiteness properties of certain subgroups of G. We consider the symmetric automorphism group ΣAut n and pure symmetric automorphism group PΣAut n of the free group F n , and inspect their BNSR-invariants. We prove that for n ≥ 2, all the "positive" and "negative" character classes of PΣAut n lie in Σ n−2 (PΣAut n ) \ Σ n−1 (PΣAut n ). We use this to prove that for n ≥ 2, Σ n−2 (ΣAut n ) equals the full character sphere S 0 of ΣAut n but Σ n−1 (ΣAut n ) is empty, so in particular the commutator subgroup ΣAut ′ n is of type F n−2 but not F n−1 . Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.

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“…Collins [4] proved that PΣ n has cohomological dimension n − 1; it also follows from his work that PΣ n is F P ∞ . Later, Brady-McCammond-Meier-Miller [1] showed that PΣ n is a duality group, and Jensen-McCammond-Meier [7] determined completely the structure of the cohomology ring of PΣ n for n ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

Finiteness properties for a subgroup of the pure symmetric automorphism group

Pettet

2009

Comptes Rendus. Mathématique

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Abstract. Let F n be the free group on n generators, and let PΣ n be the group of automorphisms of F n that send each generator to a conjugate of itself. The kernel K n of the homomorphism PΣ n → PΣ n−1 , induced by mapping one of the free group generators to the identity, is finitely generated. We show that K n has cohomological dimension n − 1, and that H i (K n ; Z) is not finitely generated for 2 ≤ i ≤ n − 1. It follows that K n is not finitely presentable for n ≥ 3.

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The Pure Symmetric Automorphisms of a Free Group Form a Duality Group

Abstract: The pure symmetric automorphism group of a finitely generated free group consists of those automorphisms which send each standard generator to a conjugate of itself. We prove that these groups are duality groups.  2001 Elsevier Science

Cited by 15 publications

References 12 publications

Sigma theory and twisted conjugacy classes

Sigma theory and twisted conjugacy classes

Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties

Finiteness properties for a subgroup of the pure symmetric automorphism group

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