Algebraic properties of mapping class groups of surfaces

N. V. Ivanov

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The present paper is the notes of a mini-course addressed mainly to non-experts. Its purpose is to provide a first approach to the theory of mapping class groups of non-orientable surfaces.

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Fox pairings and generalized Dehn twists

Gwénaël Massuyeau, Vladimir Turaev (2013)

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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.