Proper Actions of Automorphism Groups of Free Products of Finite Groups

Chen, Yuqing; Glover, Henry; Jensen, Craig A.

doi:10.1142/s0218196705002256

Cited by 4 publications

(12 citation statements)

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“…Remarque 2.1. Dans [CGJ05], on suppose dès le début que tous les groupes G i sont finis, mais cette hypothèse n'intervient pas dans la démonstration du premier théorème reformulé ci-dessus.…”

Section: (En Effet On Dispose D'une Bijection Canoniqueunclassified

Stabilité Homologique pour les Groupes d’automorphismes des Produits Libres

Collinet¹,

Djament²,

Griffin³

2012

International Mathematics Research Notices

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On montre dans cet article que, pour tout groupe G indécomposable pour le produit libre * et non isomorphe à Z, l'inclusion canonique Aut(G * n ) → Aut(G * n+1 ) induit un isomorphisme entre les groupes d'homologie Hi pour n ≥ 2i + 2, comme l'avaient conjecturé Hatcher et Wahl. En fait, on montre un peu plus -en particulier, le résultat vaut pour tout groupe G à condition de remplacer le groupe des automorphismes du produit libre par le sous-groupe des automorphismes symétriques. Nous nous appuyons pour cela sur des constructions et résultats d'acyclicité dus à McCullough-Miller et Chen-Glover-Jensen et sur des propriétés de fonctorialité qui nous permettent d'utiliser des méthodes classiques d'homologie des foncteurs. AbstractWe show in this article that, for any group G indecomposable for the free product * and non-isomorphic to Z, the canonical inclusion Aut(G * n ) → Aut(G * n+1 ) induces an isomorphism between the homology groups Hi for n ≥ 2i + 2, as was conjectured by Hatcher and Wahl. In fact we show a little more -in particular, the result is true for any group G if we replace the automorphism group of the free product by the subgroup of symmetric automorphisms. For this purpose we use constructions and acyclicity results due to McCullough-Miller and Chen-Glover-Jensen and functoriality properties which allow us to apply classical methods in functor homology. théorème de F. Laudenbach montre que, pour V n = (S 2 × S 1 ) ♯n , l'application M CG(V n ) → Aut(π 1 (V n )) = Aut(Z * n ) est un épimorphisme dont le noyau est le sous-groupe distingué engendré par les twists de Dehn autour de copies de S 2 dans V n (la notation M CG(V ) désignant le groupe des difféotopies de V ). Dans [HW10], Hatcher et Wahl montrent des résultats plus généraux sur l'homologie des groupes de difféotopies, et en déduisent que α G n,i est un isomorphisme dès que n ≥ 2i + 2 pour G = Z/n avec n ∈ {2, 3, 4, 6} et pour G = π 1 (S) avec S une surface orientable compacte sans bord, ou S une variété hyperbolique de dimension 3, de volume fini, n'admettant pas d'isométrie inversant l'orientation. Ces auteurs conjecturent alors que α G n,i est un isomorphisme pour n ≥ 2i + 2 quel que soit le groupe G.Dans cet article, nous démontrons le résultat suivant.

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Section: (En Effet On Dispose D'une Bijection Canoniqueunclassified

Stabilité Homologique pour les Groupes d’automorphismes des Produits Libres

Collinet¹,

Djament²,

Griffin³

2012

International Mathematics Research Notices

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“…Their work complements the results of Hatcher [Hat95] for G ∼ = Z and extends results of Hatcher-Wahl [HW10] for several important classes of groups G coming from low-dimensional topology. Collinet-Djament-Griffin prove their results using the theory of functor homology, and an analysis of the action of FR(G * n ) on a variation of the MacCullough-Miller complex [MM96] due to Chen-Glover-Jensen [CGJ05].…”

Section: Symmetric Automorphisms Of Free Productsmentioning

confidence: 92%

$$\text {FI}_{\mathcal {W}}$$ FI W -modules and constraints on classical Weyl group characters

Wilson

2015

Math. Z.

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In this paper we study the characters of sequences of representations of any of the three families of classical Weyl groups W n : the symmetric groups, the signed permutation groups (hyperoctahedral groups), or the even-signed permutation groups. Our results extend work of Church, Ellenberg, Farb, and Nagpal [CEF12], [CEFN14] on the symmetric groups. We use the concept of an FI W -module, an algebraic object that encodes the data of a sequence of W nrepresentations with maps between them, defined in the author's recent work [Wil14].We show that if a sequence {V n } of W n -representations has the structure of a finitely generated FI W -module, then there are substantial constraints on the growth of the sequence and the structure of the characters: for n large, the dimension of V n is equal to a polynomial in n, and the characters of V n are given by a character polynomial in signed-cycle-counting class functions, independent of n. We determine bounds the degrees of these polynomials.We continue to develop the theory of FI W -modules, and we apply this theory to obtain new results about a number of sequences associated to the classical Weyl groups: the cohomology of complements of classical Coxeter hyperplane arrangements, and the cohomology of the pure string motion groups (the groups of symmetric automorphisms of the free group).

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“…Finally, to see that (Γ Fn , γ Fn ) is proper using Proposition 3.1, we must show that, for each U ∈ Γ Fn , the maximal faces U − μ −1 U ( − → ij ) are maximal subsets of U in Γ Fn . So suppose that 7) and so by the underset condition…”

Section: The Diagonal Complex γ Fnmentioning

confidence: 99%

“…In [7], a variant of McCullough-Miller space was studied. In this variant, the bipartite trees had a chosen basepoint * .…”

Section: Based Mccullough-miller Spacementioning

confidence: 99%

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Diagonal complexes and the integral homology of the automorphism group of a free product

Griffin

2012

Proceedings of the London Mathematical Society

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Abstract. The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product.We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of diagonal complexes is introduced. This offers a generalisation of the theory of right-angled Artin groups in that each diagonal complex defines what we call a diagonal right-angled Artin group (DRAAG).

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Proper Actions of Automorphism Groups of Free Products of Finite Groups

Cited by 4 publications

References 10 publications

Stabilité Homologique pour les Groupes d’automorphismes des Produits Libres

Stabilité Homologique pour les Groupes d’automorphismes des Produits Libres

$$\text {FI}_{\mathcal {W}}$$ FI W -modules and constraints on classical Weyl group characters

Diagonal complexes and the integral homology of the automorphism group of a free product

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