In his thesis, Bieberbach solved  [Hilbert 18 problem](https://en.wikipedia.org/wiki/Hilbert%27s_eighteenth_problem) and
proved that any compact, flat Riemannian manifold is a 
quotient of a torus. I need a reference to an orbifold version
of this result: any compact, flat Riemannian manifold $M$ is a 
quotient of a torus.


It should not be hard to prove: we should take the development
map and it should give a local isometry from the orbifold
universal cover of $M$ to ${\Bbb R}^n$. The corresponding
monodromy action defines a homomorphism from the orbifold
fundamental group of $M$ to the group of affine isometries.
The rotational part of its image is finite by Margulis lemma.


However, I am pretty sure it's published somewhere,
and it's always safer (and more ethical) to cite.


Thanks in advance.



Bieberbach theorem for compact, flat Riemannian orbifolds
Bieberbach‘s 1911-12 paper ([part 1](https://gdz.sub.uni-goettingen.de/id/PPN235181684_0070?tify=%7B%22pages%22:[313],%22view%22:%22thumbnails%22%7D), [part2](https://gdz.sub.uni-goettingen.de/id/PPN235181684_0072?tify=%7B%22pages%22:[414],%22view%22:%22thumbnails%22%7D)) proves a result about groups rather than manifolds, and it does not assume the groups to be torsion-free. In today’s language it says that a discrete, cocompact group of Euclidean isometries contains its subgroup of translations (which is necessarily a free Abelian group) as a subgroup of finite index. So you can just cite Bieberbach.


