Twin groups representations
Mohamad N. Nasser
TL;DR
The paper tackles representations of the twin group $T_n$ and its extensions $VT_n$ and $WT_n$ by constructing two distinct representations: $η_1: T_n \to Aut(\mathbb{F}_n)$ via explicit automorphisms and its Magnus–Fox matrix realization in $GL_n(\mathbb{Z}[t^{\pm1}])$, and $η_2: T_n \to GL_n(\mathbb{Z}[t^{\pm1}])$ as a family of block matrices. It analyzes irreducibility of the composition factor $η_1'$ and establishes faithfulness for $n=2,3$, while showing $η_2$ is faithful only for $n=2$ and unfaithful for $n\ge 3$. The work further studies $2$-local extendability: both representations admit $2$-local extensions to $VT_n$ for all $n\ge 2$, but $η_1$ has no $2$-local extension to $WT_n$ for $n\ge 3$, whereas $η_2$ does extend to $WT_n$ (and its $2$-local extensions are described). These results contribute to understanding representations of twin-group variants and the structure of their extensions, with potential implications for related configuration-space problems and braid-group analogues.
Abstract
We construct two representations of the twin group $T_n, n\geq 2$, namely $η_1: T_n \rightarrow \text{Aut}(\mathbb{F}_n)$ and $η_2: T_n \rightarrow \text{GL}_n(\mathbb{Z}[t^{\pm 1}])$, where $\mathbb{F}_n$ is a free group with $n$ generators and $t$ is indeterminate. We then analyze some characteristics of these two representations, such as irreducibility and faithfulness. Moreover, we prove that both representations can be extended to the virtual twin group $VT_n$ in the $2$-local extension way, for $n\geq 2$, and we find their $2$-local extensions. On the other hand, we obtain a different result for the welded twin group $WT_n$. More deeply, we show that $η_1$ cannot be extended to $WT_n$ in the $2$-local extension way, for $n\geq 3$, while $η_2$ can be extended to $WT_n$ in the $2$-local extension way, for $n\geq 2$, and we find its $2$-local extensions.