Fox pairings and generalized Dehn twists

Massuyeau, Gwénaël; Turaev, Vladimir

Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2403-2456.

Nous introduisons la notion de “forme de Fox” sur une algèbre de groupe et nous utilisons les formes de Fox pour définir des automorphismes des complétés de Malcev de groupes. Ces automorphismes étendent au cadre algébrique l’action des twists de Dehn sur les algèbres de groupes fondamentaux de surfaces. Ce travail s’inspire de la généralisation des twists de Dehn par Kawazumi–Kuno aux courbes fermées non-simples sur les surfaces.

We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawazumi–Kuno generalization of the Dehn twists to non-simple closed curves on surfaces.

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