Unrestricted virtual braids and crystallographic braid groups (Q2172333)

The group of unrestricted virtual braids, denoted by \(UVB_n\) for \(n\ge 1\), was introduced by \textit{L. H. Kauffman} and \textit{S. Lambropoulou} [J. Knot Theory Ramifications 15, No. 6, 773--811 (2006; Zbl 1105.57002)] as the analogue of fused links in the setting of braids. Such links are distinguished by their virtual linking number, are considered by \textit{T. Nasybullov} [J. Knot Theory Ramifications 25, No. 14, Article ID 1650076, 21 p. (2016; Zbl 1383.57012)] as the closure of unrestricted virtual braids and classified in terms of Gauss diagrams by \textit{B. Audoux} et al. [Mich. Math. J. 67, No. 3, 647--672 (2018; Zbl 1406.57003)]. Those groups may be decomposed as a semi-direct product of a right-angled Artin group, which is in fact, by \textit{V. G. Bardakov} et al. [J. Knot Theory Ramifications 24, No. 12, Article ID 1550063, 23 p. (2015; Zbl 1360.20026)], the pure subgroup \(UVP_n\) of \(UVB_n\), by the symmetric group \(S_n\). \par The main aim of this paper is to characterise the torsion elements of \(UVB_n\) using this decomposition, namely to show that any element of finite order is a conjugate of an element of \(S_n\) by an element of \(UVP_n\). The authors show that the crystallographic braid group \(B_n=B_n/[P_n,P_n]\) embeds naturally in the group of unrestricted virtual braids \(UVB_n\), Then, they give new proofs of known results by \textit{D. L. Gonçalves} et al. [J. Algebra 474, 393--423 (2017; Zbl 1367.20034)] about the torsion elements of \(B_n=B_n/[P_n,P_n]\), and characterise the torsion elements of \(UVB_n\), where \(B_n\) is the Artin braid group, \(P_n\) is the pure Artin braid group, and \([P_n,P_n]\) is its commutator subgroup.