Question 1. Let $G$ be a subgroup of isometries of $\mathbb{R}^n$ that acts discretely and cocompactly (i.e., $\mathbb{R}^n/G$ is compact). Then there exists a finite index subgroup $\mathbb{Z}^n\subset G$.
This appears as the final part of the Cheeger–Gromoll structure theorem for nonnegative Ricci curvature, Theorem 7.3.11 in Riemannian Geometry (Third Edition) by Peter Petersen. The proof there goes like this: Let $\mathbb{R}^n$ be the normal subgroup of translations. The author says that $G\cap\mathbb{R}^n$ is a finitely generated abelian group with finite index in $G$ that acts discretely and cocompactly on $\mathbb{R}^n$. However, I don't see why this is true. In fact, I think $G\cap\mathbb{R}^n$ may well be the identity.
Question 2. If moreover $G$ acts freely, then $G=\mathbb{Z}^n$ and $\mathbb{R}^n/G$ is a flat torus.
Again this is content of Corollary 7.3.15 in that book, whose proof I cannot understand.
So how do I prove the above facts? And by the way, are there any books on discrete isometry groups?
Edit: There is something wrong about the second statement. It is required that the first Betti number of $\mathbb{R}^n/G$ should be $n$. (Otherwise you could have things like the Klein bottle.) Then the proof in the book works.
