Farrell–Zdravkovska conjecture on geometric bounding of flat and almost-flat manifolds

Let $M^n$ be a compact, connected Riemannian manifold. In case (a), $M^n$ is flat; in case (b), $M^n$ is almost flat. A manifold bounds geometrically when it is the boundary of a compact manifold whose interior supports the specified complete finite-volume negatively curved metric.

Farrell–Zdravkovska conjecture.

(a) If $M^n$ is a flat Riemannian manifold, then

$$M^n=\partial W^{n+1},$$

where $W\setminus\partial W$ supports a complete hyperbolic structure with finite volume.

(b) If $M^n$ is an almost-flat Riemannian manifold, then

$$M^n=\partial W^{n+1},$$

where $W\setminus\partial W$ supports a complete Riemannian metric with finite volume and negative sectional curvatures.

The conjecture asserts that every compact connected flat manifold and every compact connected almost-flat manifold geometrically bounds in the corresponding sense. The source attributes both statements to Farrell and Zdravkovska; their resolution status is not specified here.