Futer–Wise cubulation approximation conjecture for hyperbolic manifolds

Let $M= obreak\backslash \mathbb{H}^n$ be a closed hyperbolic $n$-manifold with fundamental group $\Gamma$, where either $n\leq 3$ or $M$ is arithmetic of simplest type. A $\lambda$-quasi-isometry is a $\Gamma$-equivariant map satisfying the quasi-isometry inequalities with multiplicative distortion $\lambda>1$. Futer–Wise conjecture. For every $\lambda>1$, $M$ is homotopy equivalent to a compact non-positively curved cube complex $\mathcal{X}$ such that there is a $\Gamma$-equivariant $\lambda$-quasi-isometry from $\widetilde{\mathcal{X}}$ to $\widetilde{M}$. The conjecture predicts that there are enough cubulations to approximate geometric actions on $\mathbb{H}^n$. It cannot be strengthened to require $1$-quasi-isometries, because translation lengths of geometric cubical actions have discrete constraints whereas the marked length spectrum of a uniform hyperbolic lattice is not contained in a discrete subgroup of $\mathbb{R}$.