TITLE: When is a Baumslag-Solitar group linear? 
QUESTION [10 upvotes]: The Baumslag-Solitar group $BS(m,n)$ is given by the group presentation 
$BS(m,n)=(a,b|ba^{m}b^{-1}=a^{n})$. When does it embed into a linear group? Thanks!

REPLY [11 votes]: The metabelian groups $BS(n,1)\simeq BS(1,n)=\langle a,b\mid aba^{-1}=b^n \rangle$ are also linear (this seems not mentionened in the Wikipedia article). A faithful, linear representation $BS(1,n)\hookrightarrow GL_2(\mathbb{R})$ is given
by 
$$
a\mapsto \begin{pmatrix} n^{\frac{1}{2}} & 0 \cr 
0 & n^{-\frac{1}{2}} \end{pmatrix}, \quad
b\mapsto \begin{pmatrix} 1 & 1 \cr
0 & 1 \end{pmatrix}.
$$
This representation is not discrete, and the groups are not polycyclic (except for $n=\pm 1$).