It is a theorem of Wolfgang Lück that a homomorphism $\varphi \colon G_0(\mathbb Z \Gamma) \to \mathbb Z$ can be constructed with the property $\varphi([\mathbb Z \Gamma]) = 1$ if $\Gamma$ is amenable. I Moreover, such a homomorphism cannot exist if $\Gamma$ contains a non-abelian free group.
It is conjectured that the existence is a characterization of amenability.