Fibration conjecture for compact Clifford–Klein forms

Let $G/H$ be a reductive homogeneous space with $G$ and $H$ connected. Let $L$ be a maximal compact subgroup of $H$, and let $K$ be a maximal compact subgroup of $G$ containing $L$. Let $\Gamma$ be a torsion-free discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. Then there exists a closed manifold $B$ of dimension $p$ such that

• $\pi_1(B)$ is isomorphic to $\Gamma$;
• the universal cover of $B$ is contractible;
• $\Gamma\backslash G/H$ admits a fibration over $B$ with fibers of the form $gK/L$ for some $g\in G$.

Fibration conjecture. Under these hypotheses, such a manifold $B$ and fibration exist. This generalizes a theorem of Guérithaud–Kassel for certain quotients of $\operatorname{SO}_0(d,1)$; the source gives no resolution of the general assertion.