I am reading de la Harpe's book "Topics in Geometric Group Theory".
On page 145, there is a theorem:
Let $V$ be a complete $n$-Riemannian manifold with sectional curvature satisfying $K\ge 0$. Then there exist a finite normal subgroup $F\_1$ of the fundamental group $\pi\_1(V)$ and a short exact sequence of the form:
$$
1 \to \mathbb Z^k \to \pi\_1(V)/F\_1 \to F\_2 \to 1
$$
where $k\le n$ and where $F\_2$ is a finite group.


I have two questions:


1) why the author put $F\_1$ in such a special position, why not absorb $F\_1$ into $F\_2$, since they are basically all finite group.


2) In other papers/books, peopole called a discrete, torsion free and cocompact subgroup of $O(n)\ltimes \mathbb R^n $ A Bieberbach group. So why "torsion free" is important?


Thank you!



Extension of groups in Bieberbach's theorem
There is a ``quasi-isometric rigidity'' context in which a finite kernel arises naturally. I am not sure how general this is so let me state it in the narrower context of constant sectional curvature equal to zero. 


Theorem (quasi-isometric extension of the Bieberbach Theorem): If $\Gamma$ is an abstract group quasi-isometric to Euclidean $n$-space then there exists a finite normal subgroup $F\_1$ of $\Gamma$ such that $\Gamma / F\_1$ is a Bieberbach group (of the general kind, allowing torsion), and there is a short exact sequence of the form $1 \to \mathbb{Z}^n \to \Gamma / F\_1 \to F\_2 \to 1$ where $F\_2$ is a finite group. So the group $\Gamma/F\_1$ is the orbifold fundamental group of a compact $n$-dimensional Euclidean orbifold.


This is a consequence of Gromov's polynomial growth theorem. I don't know whether this is what de la Harpe had in mind, but maybe.


