I am trying to figure out why the following statement from my lecture notes is true and would appreciate any help; a statement in my lecture notes in Riemann surface stated that "Any subgroup $\Gamma$ of $\text{Aut}(\mathbb{C})$ acting properly discontinuously and freely on $\mathbb{C}$ must be trivial or conjugate to $\mathbb{Z}$ or to a lattice $\mathbb{Z} + \tau \mathbb{Z}$."
I can see that the statement holds true for the first 2 cases where $\Gamma$ is trivial and single-generated respectively. How do I show the last case?