Permutational wreath pullbacks and framed braid-type groups

TL;DR

This paper introduces permutational wreath pullbacks, studies their structure, and applies them to framed braid groups, revealing new properties and unifying various braid group types.

Contribution

It defines permutational wreath pullbacks, analyzes their structure, and applies these concepts to framed braid groups, providing new insights and unifying existing frameworks.

Findings

Determined the center and abelianization of permutational wreath pullbacks.

Established criteria for the $R_inite$-property inheritance.

Provided applications to classical, surface, virtual, and singular framed braid groups.

Abstract

Let $\sigma\colon G \to S_n$ be a surjective homomorphism and let $H$ be a group. We introduce the \emph{permutational wreath pullback} \[ H \wr_\sigma G = H^n \rtimes_\sigma G, \] where the action of $G$ on $H^n$ is induced by permutation of coordinates via $\sigma$, and undertake a systematic structural study of this construction. We determine the center and the abelianization in full generality. We further show that $H \wr_\sigma G$ admits a natural interpretation as the pullback of the classical wreath product $H \wr S_n$ along $\sigma$, providing a conceptual explanation for its functorial behavior. When $H$ is finitely generated abelian, we establish a criterion for the abelian kernel $H^n$ to be characteristic and for $H \wr_\sigma G$ to inherit the $R_\infty$-property from $G$; we verify this criterion for kernels arising from the virtual braid group $VB_n$ and the virtual twin group $VT_n$, obtaining new families of framed groups with the $R_\infty$-property. Rigidity results show that the abelian kernel, $n$, $H$, and $G$ are determined by the abstract group $H \wr_\sigma G$. 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.