Realisability of $G_{n}^{3}$, realisability projection, and kernel of the $G_{n}^{3}$-braid presentation

TL;DR

This paper proves that the kernel of a specific group homomorphism from the pure braid group to G_{n}^{3} is generated by full twist braids and their exponents, using realisability projections and induction.

Contribution

It introduces the concept of realisability projection and proves the kernel characterization for the map from pure braid groups to G_{n}^{3}.

Findings

Kernel consists of full twist braids and their exponents.

Realisability projection ensures equivalence of realisable G_{4}^{3}-elements.

Inductive proof extends results to all n ≥ 4.

Abstract

The aim of this article is to prove that the kernel of the map from the pure braid group $PB_{n},n\ge 4$ to the group $G_{n}^{3}$ consists of full twist braids and their exponents. The proof consists of two parts. The first part which deals with $n=4$ relies on the crucial tool in this construction having its own interest is the {\em realisability projection} saying that if two {\em realisable} $G_{4}^{3}$-elements are equivalent then they are equivalent by a sequence of realisable ones. The second part (an easy one) uses induction on $n$.