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

The group of unimodular automorphisms of $\mathbb{C}^2$ is hopfian