Group action of the Baumslag-Solitar groups

Question

The Baumslag-Solitar groups are defined by

$$G=BS(m,n)=\langle a,b: ba^{m}b^{-1}=a^{n}\rangle\,,$$

where $m,n$ are integers.

My question is: Is there a linear action of $G=BS(1,2)$ over $\mathbb{R}^{2}$ ?

Answer 1

Yes, the matrices $\begin{bmatrix}2&0\\0&1\end{bmatrix}$ and $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ generate a copy of $BS(1,2)$. So this gives an action on $\mathbb{R}^2$.