Irreducible Representations of Baumslag-Solitar Groups

We classify the finite-dimensional irreducible linear representations of the Baumslag-Solitar groups BS(p,q) = < a, b | a b^p = b^q a > for relatively prime p and q. The general strategy of the argument is to consider the matrix group given by image of a representation and study its Zariski closure in GL(n, C).