Twin groups representations

TL;DR

This paper constructs and analyzes two representations of the twin group, exploring their properties, extensions to the virtual twin group, and differences in extension possibilities for the welded twin group.

Contribution

It introduces two specific representations of the twin group, studies their properties, and examines their extension capabilities to related groups, revealing new structural insights.

Findings

Both representations can be extended to the virtual twin group in the 2-local way.

η_1 cannot be extended to WT_n in the 2-local way for n≥3.

η_2 can be extended to WT_n in the 2-local way for n≥2.

Abstract

We construct two representations of the twin group $T_n, n\geq 2$, namely $\eta_1: T_n \rightarrow \text{Aut}(\mathbb{F}_n)$ and $\eta_2: T_n \rightarrow \text{GL}_n(\mathbb{Z}[t^{\pm 1}])$, 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 $\eta_1$ cannot be extended to $WT_n$ in the $2$-local extension way, for $n\geq 3$, while $\eta_2$ can be extended to $WT_n$ in the $2$-local extension way, for $n\geq 2$, and we find its $2$-local extensions.