2026/04/07 by Ênio Leite, Oscar Ocampo · 1 voice
Mathematics · #math.GR #math.GT
Let σ\colon G → Sn be a surjective homomorphism and let H be a group. We introduce the permutational wreath pullback H \wrσG = Hn \rtimesσG, where the action of G on Hn is induced by permutation of coordinates via σ, and undertake a systematic structural study of this construction. We determine the center and the abelianization in full generality. We further show that H \wrσG admits a natural interpretation as the pullback of the classical wreath product H \wr Sn along σ, providing a conceptual explanation for its functorial behavior. When H is finitely generated abelian, we establish a criterion for the abelian kernel Hn to be characteristic and for H \wrσG to inherit the R_∞-property from G; we verify this criterion for kernels arising from the virtual braid group VBn and the virtual twin group VTn, obtaining new families of framed groups with the R_∞-property. Rigidity results show that the abelian kernel, n, H, and G are determined by the abstract group H \wrσG. Applications include uniform descriptions of classical, surface, virtual, and singular framed braid groups, and a reduction of splitting problems for framed surface braid groups to the classical Fadell--Neuwirth setting.