Two-generator one-relator groups and marked polytopes
Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 831-879.

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.

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.

Received: 2016-08-03
Revised: 2017-09-20
Accepted: 2018-02-12
Published online: 2020-05-28
DOI: https://doi.org/10.5802/aif.3325
Classification: 20J05,  20F65,  22E40,  57R19
Keywords: Finitely presented group, Novikov ring, BNS invariant, Sigma invariant, Fox calculus
@article{AIF_2020__70_2_831_0,
     author = {Friedl, Stefan and Tillmann, Stephan},
     title = {Two-generator one-relator groups and marked polytopes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {2},
     year = {2020},
     pages = {831-879},
     doi = {10.5802/aif.3325},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_2_831_0/}
}
Friedl, Stefan; Tillmann, Stephan. Two-generator one-relator groups and marked polytopes. Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 831-879. doi : 10.5802/aif.3325. https://aif.centre-mersenne.org/item/AIF_2020__70_2_831_0/

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