Quotient characterization of compact integral-integral affine manifolds

Establish whether every compact integral-integral affine manifold is the quotient of R^n by a free and proper action of a discrete group of integral-integral affine transformations of the form x ↦ Ax + b with A ∈ GL_n(Z) and b ∈ Z^n.

Background

The paper introduces integral-integral affine manifolds—manifolds equipped with atlases whose transition maps are affine with integer linear and integer translational parts—and studies their geometry in connection with Lagrangian torus fibrations and geometric quantization.

Typical examples arise as quotients R^n/Γ where Γ is a group of integral-integral affine transformations acting freely and properly, such as tori, Klein bottles, and Kodaira–Thurston-type manifolds.

The authors explicitly pose whether every compact integral-integral affine manifold is of this quotient form. They note that this question appears to be open and relate it to the Markus conjecture, which predicts geodesic completeness under det(A)=±1 transition maps for closed affine manifolds.