Arithmetic topology in Ihara theory

TL;DR

This paper explores the analogies between braid group invariants and arithmetic Galois representations within Ihara theory, advancing understanding of their interrelations and potential applications.

Contribution

It develops new connections between braid invariants and arithmetic Galois representations, extending Ihara theory's framework.

Findings

Identifies relations between Milnor invariants and Galois representations

Establishes links between Johnson homomorphisms and arithmetic structures

Analyzes Magnus-Gassner cocycles in the context of number theory

Abstract

Ihara initiated to study a certain Galois representation which may be seen as an arithmetic analogue of the Artin representation of a pure braid group. We pursue the analogies in Ihara theory further, following after some issues and their inter-relations in the theory of braids and links such as Milnor invariants, Johnson homomorphisms, Magnus-Gassner cocycles and Alexander invariants, and study relations with arithmetic in Ihara theory.