Fox pairings and generalized Dehn twists

TL;DR

This paper introduces Fox pairings in group algebras to define automorphisms of Malcev completions, generalizing Dehn twists algebraically and inspired by their extension to non-simple curves on surfaces.

Contribution

It develops a new algebraic framework using Fox pairings to generalize Dehn twists in the context of group algebra automorphisms.

Findings

Defined Fox pairings in group algebras.

Constructed automorphisms of Malcev completions.

Connected algebraic automorphisms to geometric Dehn twists.

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.