Unrestricted virtual braids and crystallographic braid groups

TL;DR

This paper explores the algebraic structure of crystallographic braid groups and unrestricted virtual braid groups, establishing embeddings, analyzing torsion elements, and providing new proofs for known properties.

Contribution

It introduces an embedding of crystallographic braid groups into unrestricted virtual braid groups and characterizes the torsion elements of the latter, offering new insights and proofs.

Findings

Crystallographic braid group embeds in unrestricted virtual braid group.

Torsion elements of $B_n/[P_n,P_n]$ are characterized.

Torsion elements of $UVB_n$ are fully characterized.

Abstract

We show that the crystallographic braid group $B_n/[P_n,P_n]$ embeds naturally in the group of unrestricted virtual braids $UVB_n$, we give new proofs of known results about the torsion elements of $B_n/[P_n,P_n]$, and we characterise the torsion elements of $UVB_n$.