no. 2
Two-generator one-relator groups and marked polytopes
[Les groupes à deux générateurs et une relation et les polytopes marqués]
Friedl, Stefan1 ; Tillmann, Stephan2
1 Fakultät für Mathematik Universität Regensburg 93053 Regensburg, Germany
2 School of Mathematics and Statistics The University of Sydney NSW 2006 Sydney (Australia)
Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 831-879.

Nous utilisons le calcul de Fox pour attribuer un polytope marqué à presque chaque groupe à deux générateurs et une relation. En reliant les sommets marqués à l’homologie de Nivokov-Sikorav, nous démontrons que le polytope marqué détermine l’invariant de Bieri–Neumann–Strebel du groupe. De plus nous démontrons que très souvent le polytope marqué est un invariant du groupe et que si c’est le cas le polytope marqué détermine les complexités minimales des scindements HNN du groupe.

We use Fox calculus to assign a marked polytope to a “nice” group presentation with two generators and one relator. Relating the marked vertices to Novikov–Sikorav homology we show that they determine the Bieri–Neumann–Strebel invariant of the group. Furthermore we show that in most (possibly all) cases the marked polytope is an invariant of the underlying group and that in those cases the marked polytope also determines the minimal complexity of all the associated HNN-splittings.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3325
Classification : 20J05, 20F65, 22E40, 57R19
Keywords: Finitely presented group, Novikov ring, BNS invariant, Sigma invariant, Fox calculus
Mot clés : groupe de présentation finie, anneau de Novikov, invariant de BNS, invariant Sigma, calcul de Fox

Friedl, Stefan 1 ; Tillmann, Stephan 2

1 Fakultät für Mathematik Universität Regensburg 93053 Regensburg, Germany
2 School of Mathematics and Statistics The University of Sydney NSW 2006 Sydney (Australia)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Friedl, Stefan; Tillmann, Stephan. Two-generator one-relator groups and marked polytopes. Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 831-879. doi : 10.5802/aif.3325. https://aif.centre-mersenne.org/articles/10.5802/aif.3325/

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