Twin groups representations

Abstract

We construct two representations of the twin group $T_n,n\ge 2$, namely $\hat{I}{\cdot }_1:T_n\to \text{Aut}({\mathbb{F}}_n)$ and $\hat{I}{\cdot }_2:T_n\to {\text{GL}}_n(\mathbb{Z}[t^{\pm 1}])$, 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 $V{T}_n$ in the $2$-local extension way, for $n\ge 2$, and we find their $2$-local extensions. On the other hand, we obtain a different result for the welded twin group $W{T}_n$. More deeply, we show that $\hat{I}{\cdot }_1$ cannot be extended to $W{T}_n$ in the $2$-local extension way, for $n\ge 3$, while $\hat{I}{\cdot }_2$ can be extended to $W{T}_n$ in the $2$-local extension way, for $n\ge 2$, and we find its $2$-local extensions.