Quotient characterization of compact integral-integral affine manifolds

Establish whether every compact integral-integral affine manifold is affinely equivalent to a quotient of R^n by a free and proper action of a discrete group of integral-integral affine transformations, thereby determining the global realization of such structures via developing maps and holonomy.

Background

Integral-integral affine manifolds are defined by atlases whose transition maps are x ↦ Ax + b with A in GL_n(Z) and b in Z^n. Classical examples (tori, Klein bottles, Kodaira–Thurston-like manifolds) arise as quotients of R^n by discrete groups of integral-integral affine transformations acting freely and properly.

The authors explicitly pose the general question of whether all compact integral-integral affine manifolds admit such a quotient description. They note this appears to be open and relate it to the Markus conjecture concerning geodesic completeness of closed affine manifolds with unimodular linear parts.