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FOX PAIRINGS AND GENERALIZED DEHN TWISTS

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Niveau: Supérieur, Doctorat, Bac+8
FOX PAIRINGS AND GENERALIZED DEHN TWISTS GWENAEL MASSUYEAU AND VLADIMIR TURAEV Abstract. 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. 1. Introduction There is a simple and well-known construction producing families of automor- phisms of modules from bilinear forms. Given a module H over a commutative ring K and a bilinear form : H ? H ? K, one associates with any isotropic vector a ? H and any k ? K a transvection H ? H carrying each h ? H to h+ k(a h)a. We introduce in this paper a group-theoretic version of transvections. Note that any group pi has a Malcev completion pi = piK formed by the group-like elements of the Hopf algebra K?[pi] which is the fundamental completion of the group algebra K[pi], see [Qu]. Our main construction starts with a group pi and a certain bilinear form, a Fox pairing, in K[pi] and produces a family of group automorphisms of pi which are in many respects similar to transvections.

  • kx ?

  • any

  • fox derivative

  • fox derivative carrying

  • group pi

  • aut pi

  • dehn twist


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