17 problems

Nonexistence conjecture for $\operatorname{SO}(2,2m+1)/\operatorname{SU}(1,m)$. The homogeneous space

Converse to Theorem 15. The converse of Theorem 15 also holds.

Let $G_\mathbb{C}/K_\mathbb{C}$ be an irreducible complex symmetric space, where $G_\mathbb{C}$ is a complex semisimple Lie group and $K_\mathbb{C}$ is the fixed-point subgroup of…

Let $n$ be a positive integer, and let $SO(n+1,\mathbb{C})/SO(n,\mathbb{C})$ be the $n$-dimensional complex sphere. Kobayashi's complex sphere conjecture. The complex sphere admits…

For a homogeneous space $G/H$ of reductive type, write (P-cocH$^n$) if there exists a discrete subgroup of $G$ isomorphic to a cocompact lattice of $\operatorname{O}(n,1)$ acting p…

Let $G$ be a connected real linear reductive Lie group, let $H$ be a reductive subgroup of $G$, and let $\Gamma$ be a discrete subgroup of $G$. Let the Zariski closure of $\Gamma$…

Let $G$ be a non-compact real linear semisimple Lie group and let $H$ be a reductive subgroup. A compact Clifford–Klein form of $G/H$ is a quotient by a discrete subgroup acting fr…

The conjecture. The only compact locally homogeneous simple pseudo-Riemannian Einstein manifolds are of the form $\Gamma\backslash G/H$, determined by either some standard triple…

Let $L \backslash H$ be a homogeneous space admitting an action by left multiplications of a simple non-compact Lie group $G$, and suppose that $L$ is non-compact. Zimmer's conject…

Let $G$ be a 1-connected nilpotent Lie group, $H \subset G$ a closed subgroup, and $\Gamma \subset G$ a non-trivial discrete subgroup. Let $\mathcal{R}(\Gamma,G,H)$ denote the spac…

Fibration conjecture. Under these hypotheses, such a manifold $B$ and fibration exist. This generalizes a theorem of Guérithaud–Kassel for certain quotients of…

Let $n\geq 2$. Let $\Gamma$ be a discrete group and let $j\times\rho$ be a faithful representation of $\Gamma$ into $\operatorname{SO}_0(n,1)\times\operatorname{SO}_0(n,1)$. Suppos…

Let $n\geq 2$, let $\Gamma$ be a group, and let $j,\rho:\Gamma\to\operatorname{SO}_0(n,1)$ be representations such that $j\times\rho(\Gamma)$ acts properly discontinuously and coco…

Let $n=2k+1>4$ be odd, and let $\Gamma$ be a discrete subgroup of $\operatorname{SO}_0(n-1,2)$ acting freely, properly discontinuously and cocompactly on anti-de Sitter space…

Let $\tilde H^{1,2m}$ be the universal anti de Sitter spacetime of dimension $2m+1$, and let a compact quotient be a manifold of the form $\Gamma\setminus\tilde H^{1,2m}$. Such a q…

Let $m$ be a positive integer and set … with $H$ embedded in $G$ in the standard way. A compact form of $H\backslash G$ is a compact quotient by a properly discontinuous group acti…

Sharpness conjecture. Any compact Clifford–Klein form of $X=G/H$ is sharp.