  Let $G$ be the group of unimodular automorphisms of $\mathbb C^2$. In the
paper we prove two interesting results about this group. The first one is about
absence of non-trivial finite-dimensional representations of $G$. The second
one, we show that any non-trivial group endomorphism of $G$ is a monomorphism,
which implies that $G$ is hopfian.
