Does every finitely generated free solvable group embed into the group of polynomial automorphisms of some C^n?
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Wilhelm Magnus (W. Magnus, Über $n$dimensionale Gittertransformationen. Acta Math. 64 (1935), no. 1, 353367.) seems to have shown that the free metabelian group on $n$ generators has a faithful representation of degree $2$. It follows that there is a copy of the free solvable group on $n$ generators and of derived length $2$ inside $GL_2(\mathbb{C})$. That answers your question affirmatively in a special case. 

