2024/03/19 by Karel Dekimpe, Dekimpe, Karel, Daciberg Lima Gonçalves +3
Mathematics · #20C30 #20E45 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Primary: 20E36 #Secondary: 20F36
paper · pdf · doi:10.48550/arxiv.2403.12528
openalex publication_date 2024/03/19 · openalex created_date 2024/03/21 · openalex updated_date 2026/07/28
Let n≥ 2. Let VBn (resp. VPn) denote the virtual braid group (resp. virtual pure braid group), let WBn (resp. WPn) denote the welded braid group (resp. welded pure braid group) and let UVBn (resp. UVPn) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for n≥ 4, the group VPn and for n≥ 3 the groups WPn and UVPn are characteristic subgroups of VBn, WBn and UVBn, respectively. In the second part of the paper we show that, for n≥ 2, the virtual braid group VBn, the unrestricted virtual pure braid group UVPn, and the unrestricted virtual braid group UVBn have the R_∞-property. As a consequence of the technique used for few strings we also prove that, for n=2,3,4, the welded braid group WBn has the R_∞-property and that for n=2 the corresponding pure braid groups have the R_∞-property. On the other hand for n≥ 3 it is unknown if the R_∞-property holds or not for the virtual pure braid group VPn and the welded pure braid group WPn.