I consider the Möbius transformations with fixed points $0$ and $1$ and determinant $1$. This gives effectively matrices of the form $$A_{\tau}:=\begin{pmatrix}\frac{1}{\tau} & 0 \\ \frac{1}{\tau}-\tau & \tau\end{pmatrix}$$ for some complex $\tau\neq 0$. Now, the set $\Gamma:=\lbrace A_{\tau}| \tau\in\mathbb{C}\backslash\lbrace 0\rbrace\rbrace$ induces a subgroup of $SL(2,\mathbb{C})$ and because $SL(2,\mathbb{C})$ is a Lie group (with $\mathrm{dim}_{\mathbb{C}} SL(2,\mathbb{C}) = 3$), I think that $\Gamma$ can be seen as a submanifold with complex dimension $1$ in the manifold of $SL(2,\mathbb{C})$.
But there should be more to say about it, due to the specific form of the matrices $A_{\tau}$. But I am a bit lost here. I try to find a characterization of $\Gamma$ which as complete as possible and so all ideas are welcome. Complete proofs are not necessary, I will try to do them myself. Literature recommendations are also highly appreciate. Does the subgroup induced by $\Gamma$ maybe have a specific name?