The finite subgroups of ${ \rm PGL}_2(\mathbb{ C})$ are shown to be the only finite groups $G$ with this property: for some integer $r_0$ (depending on $G$), all Galois covers $X\rightarrow \mathbb{ P}^1_{ \mathbb{ C}}$ of group $G$ can be obtained by pulling back those with at most $r_0$ branch points along non-constant rational maps $\mathbb{ P}^1_{ \mathbb{ C}} \rightarrow \mathbb{ P}^1_{ \mathbb{ C}}$. For $G\subset { \rm PGL}_2(\mathbb{ C})$, it is in fact enough to pull back one well-chosen cover with at most $3$ branch points. A consequence of the converse for inverse Galois theory is that, for $G\not \subset { \rm PGL}_2({ \mathbb{ C}})$, letting the branch point number grow provides truly new Galois realizations $F/{ \mathbb{ C}}(T)$ of $G$. Another application is that the ``Beckmann--Black'' property that ``any two Galois covers of $\mathbb{ P}^1_{ \mathbb{ C}}$ with the same group $G$ are always pullbacks of another Galois cover of group $G$'' only holds if $G\subset { \rm PGL}_2({ \mathbb{ C}})$.
Categories: math.NT