Let $G$ be a subgroup of the group of invertible affine transformations of $\mathbb{R}^n$. It is given that $G$ acts freely and properly discontinuously on $\mathbb{R}^n$. Must the Euler characterisitc of the quotient of $\mathbb{R}^n$ by $G$ be zero ?
Edit: Properly discontinuously means: for every $x\in \mathbb{R}^n$, there exists an open set $V$ containing $x$ such that for every nonidentity $g\in G$ we have $gV\cap V=\emptyset$
My question seems to be related to the Chern conjecture , but I am not knowledgeable enough to judge if the two questions are equivalent or not.
It is not equivelant to the Chern conjecture, which deals with affine manifolds.
The condition of being a quotient of $\mathbb{R}^n$ by affine maps is equivelant to being a complete affine manifold. In this case the quotient has Euler characteristic zero. This was proved by Konstant and Sullivan in 1975 (see the History part of https://en.wikipedia.org/wiki/Chern%27s_conjecture_(affine_geometry)).