Characteristic subgroups and the R∞-property for virtual braid groups

Let n >= 2. Let V B-n , (resp. V P-n) 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 UV B-n , (resp. UV P-n) 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 V P, , and for n >= 3 the groups WPn and UV P-n are characteristic subgroups of V B-n , WB(n )and UV B-n , respectively. In the second part of the paper we show that, for n >= 2, the virtual braid group V B-n , the unrestricted virtual pure braid group UV P-n , and the unrestricted virtual braid group UV B-n , have the R-infinity-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-infinity-property and that for n = 2 the corresponding pure braid groups have the R-infinity-property. On the other hand for n >= 3 it is unknown if the R-infinity-property holds or not for the virtual pure braid group V P-n and the welded pure braid group WPn . (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.