22 problems

Let $G/H$ be a homogeneous space in the reductive setting, and let $K$ be a maximal compact subgroup of $G$. Rank inequality conjecture. If $G/H$ admits a cocompact discontinuous g…

Schwartz's conjectural picture. A $(p_1,p_2,p_3)$-representation is a discrete embedding if and only if neither $w_A$ nor $w_B$ is elliptic. The corresponding parameter values form…

Let $(p,q,r)$ be a triple with $p\leq q\leq r$, let $\rho_t(p,q,r)$ be the associated family, and classify the triple as type A or type B according to the critical-interval definit…

Let $\rho_t(p,q,r)$ be the one-parameter family of representations of the $(p,q,r)$-reflection triangle group by complex reflections, with $p\leq q\leq r$ and $\rho_0$ stabilizing…

The $n$-$(n+1)$-$(n+2)$ conjecture. If $N_{1}$ is of type $\operatorname F_n$, $G_1$ and $G_2$ are of type $\operatorname F_{n+1}$, and $Q$ is of type $\operatorname F_{n+2}$, then…

Let $2\leqslant k\leqslant m$ be integers. For groups $G_{1},\ldots,G_{m}$, a subgroup $P\leqslant G_{1}\times\cdots\times G_{m}$ virtually surjects on $k$ factors if, for every…

Let $G/P$ be a flag manifold, where $G/P$ is neither real projective space nor the conformal sphere. A proper divisible domain is a proper domain in $G/P$ admitting a properly disc…

Schwartz's conjecture. The $(p,q,r;\alpha)$ representation of $\Delta_{p,q,r}$ is discrete and faithful if and only if $W_A$ and $W_B$ are non-elliptic. Furthermore, if $p<10$, it…

Measure-sequence characterization. The sequences of measures defining a density in this sense are precisely those satisfying

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 $G_0$ be a real linear simple Lie group of real rank at least $2$, let $K$ be a maximal compact subgroup of $G_0$, and let $\Gamma$ be a discrete subgroup of $G_0\times G_0$ ac…

A compact complete affine manifold is a compact manifold equipped with a complete affine structure. Auslander's conjecture. The fundamental group of a compact complete affine manif…

Finite presentability conjecture. Then $H$ is finitely presented.

Let $G$ be a simple Lie group with real rank at least $2$. For a countable group $\Gamma$, let $A_i$ consist of elements whose centralizer contains a free abelian subgroup of rank…

Let $G$ be a real algebraic group, let $H\subgroup G$ contain a maximal reductive subgroup of $G$, and let $\Gamma\subgroup G$. Say that $\Gamma$ acts crystallographically on $G/H$…

The short-word discreteness conjecture. A $(n_1,n_2,n_3)$-complex hyperbolic triangle group is a discrete embedding in $PU(2,1)$ if neither $W_A$ nor $W_B$ is non-elliptic.

Let $F_2$ be the free group of rank $2$, let $\operatorname{PSL}_2\mathbb{C}$ be the group of orientation-preserving isometries of hyperbolic $3$-space, and let…

Let $\mathcal{A}$ be an exact C-algebra, meaning that its reduced C-algebra is exact. Let $I(\mathcal{A})$ denote the injective envelope of $\mathcal{A}$. Ozawa's conjecture. There…

Let $\mathcal{H}$ and $\mathcal{H}^{\checkmark}$ denote the two DAHA-type algebras considered in the paper, and let $\mathcal{B}_q$ be the associated braid-group representation. Co…

Let $G$ be a Lie group and let $H$ be a reductive subgroup. A compact form of $H\backslash G$ is a compact quotient by a properly discontinuous group action. Suppose that $L$ is a…

Let $N$ be an oriented network, and call it reduced when it satisfies the paper's reducedness condition. Let the monodromy action on $N$ be the action induced by the network's mono…

Let $G$ be a non-elementary subgroup of $SL(2,\mathbb C)$ containing elliptic elements, and let $f$ be a loxodromic or elliptic transformation. For each elliptic element $g\in G$,…