Bieberbach theorem for compact, flat Riemannian orbifolds

In his thesis, Bieberbach solved Hilbert 18 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.

• If you need a textbook reference you could use "Bieberbach Groups and Flat Manifolds" by L. S. Charlap or "Spaces of constant curvature by J. A Wolf. – Igor Belegradek Feb 9 at 17:59
• does it have the result stated for orbifolds? – Misha Verbitsky Feb 9 at 18:05
• They don't use the word "orbifold". Everything is stated for discrete isometry groups of $\mathbb R^n$. Which is the same thing because flat orbifolds are good. – Igor Belegradek Feb 9 at 18:07
• It seems you are unaware of the fact that complete nonpositively curved orbifolds are good (i.e., developable). This is due to Gromov (I think) and proved e.g. in Bridson-Haefliger "Metric spaces of nonpositive curvature". – Igor Belegradek Feb 9 at 19:08
• thanks, I would look in this book – Misha Verbitsky Feb 10 at 19:05