We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi-direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi-direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as quotients the singular braid group, virtual braid group, welded braid group, and classical braid group.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-1,
author = {Valerij G. Bardakov},
title = {The virtual and universal braids},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {1-18},
zbl = {1078.20036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-1}
}
Valerij G. Bardakov. The virtual and universal braids. Fundamenta Mathematicae, Tome 184 (2004) pp. 1-18. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-1/