Structure and automorphisms of pure virtual twin groups

Virtual braid groups, virtual twin groups and crystallographic groups

“…From Remark 3.2 of [19], using the presentation of the pure virtual twin group given in Theorem 3.3, we conclude that the action by conjugation of S n = ρ 1 , . .…”

“…As mentioned at the end of the page 4 of [19], the virtual twin group admits a decomposition as semi product V T n = P V T n S n , the map ι : S n −→ V T n given by ι(τ i ) = ρ i , for i = 1, . .…”

“…We state here, without proof, the following presentation of the pure virtual twin group. Theorem 3.3 (Theorem 3.3, [19]). The pure virtual twin group P V T n on n ≥ 2 strands has the presentation • Generators:…”

“…• Relations: λ i,j λ k,l = λ k,l λ i,j for distinct integers i, j, k, l. Remark 3.4. In the presentation of P V T n given in [19] the element λ i,i+1 is equal to σ i ρ i , for i = 1, . .…”

“…The virtual twin group V T n was introduced in [2] as an abstract generalization of twin groups, together with the pure virtual twin group P V T n which is defined as the kernel of the natural surjection from V T n onto S n . Then, in [19], the virtual twin group V T n and the pure virtual twin group P V T n were studied with more details. The virtual twin group V T n contains the twin group T n and the symmetric group S n and similarly to virtual braid groups, they have a geometrical interpretation.…”