Embedding of the Group of Special Affine Transformations into $\mathrm{SL}(2,\mathbb{C})$.

Question

Is there an invective group homomorphism from the group of special affine transformations of the plane into $\mathrm{SL}(2,\mathbb{C})$? (The group of special affine transformations consists of all maps of the form $x\mapsto Ax+v$ for some $2\times 2$ matrix $A$ with $\det(A)=1$ and some vector $v\in \mathbb{R}^2$.)

Answer 1

The answer is no, because in $\mathrm{SL}_2(\mathbf{C})$ the normalizer of every non-central abelian subgroup $V$ is metabelian. Indeed we can suppose that $V$ is closed. If the normalizer is the whole group, then $V$ is normal, hence central; otherwise the normalizer is a closed proper subgroup hence is metabelian.