16 problems

Properness conjecture. The action of $\scr A(M)$ on $M$ should be proper unless $M$ is CR equivalent to the sphere or to the Heisenberg group with its standard CR structure.

Nonexistence conjecture. The homogeneous space

Standard quotient conjecture. The homogeneous space $G/H$ admits a cocompact properly discontinuous group if and only if $G/H$ admits a compact standard quotient.

Lipsman's conjecture. The following two conditions are equivalent: (i) the $L$-action on $G/H$ is proper; (ii) the $L$-action on $G/H$ has the CI property.

For $n\geqslant 2$, let $\Gamma$ be a cocompact lattice of $\operatorname{O}(n,1)$, and let $N\geqslant n$. Consider the standard embedding … A representation is strictly dominated…

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 linear real simple Lie group and let $H$ be a reductive subgroup. Write (P-surf) for the existence of a proper action of a discrete surface subgroup of genus at least…

Let $G/H$ be a homogeneous space of reductive type. Say that $G/H$ satisfies (P-free) if it admits a proper action of a non-abelian free discrete subgroup of $G$, and satisfies (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 $G/H$ be a homogeneous space of reductive type, and let $\Gamma$ be a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. Write…

Let $G$ be a locally compact group, let $K$ be a compact large subgroup of $G$, and let $S$ be a $K$-space. The neighborhood equivariant retract conjecture. $S$ is a neighborhood…

Kobayashi's special-case conjecture. This homogeneous space does not admit a cocompact discontinuous group. This is presented as a special case of the compact-standard-quotient con…

Generalized Stolz conjecture. The index map $\Theta$ is an isomorphism. This is the proper-action analogue of the Stolz conjecture, replacing $B\Gamma$ by the universal proper…

Nucinkis's conjecture. Every group of finite $\operatorname{\mathfrak{F}}$-cohomological dimension admits a finite-dimensional model for…

Invariant-metric conjecture. The topology of $X$ is metrizable by a $G$-invariant metric.

Paracompactness conjecture. If $X$ is a paracompact proper $G$-space, then the orbit space $X/G$ is paracompact.