A Group-theoretical Finiteness Theorem

We start with the universal covering space $\tilde M^n$ of a closed n-manifold and with a tree of fundamental domains which zips it $T\longarrow \tilde M^n$. Our result is that, between $T$ and $\tilde M^n$, is an intermediary object , $T\rto^p G \rTo^F \tilde M^n$, obtained by zipping, such that each fiber of p is FINITE and $T\rto^p G \rTo^F \tilde M^n$ admits a SECTION.