353 problems

Let $G$ be a Gromov hyperbolic group whose boundary at infinity is homeomorphic to the two-sphere $\mathbb{S}^2$. A group is virtually Kleinian if, up to a finite-index subgroup, i…

For every $n\ge 2$, every cocompact lattice $\Gamma<\mathrm{Isom}(\mathbb H^n)$, and every finitely supported admissible probability measure $\mu$ on $\Gamma$, the hitting measure…

For every integer $g\ge 2$, there are only finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds having a fiber of genus $g$.

For every finite-volume cusped hyperbolic $3$-manifold $M$, there exists a finite ideal triangulation of $M$ whose tetrahedra are realized as positively oriented geodesic ideal tet…

For every $N\ge 2$ and every finitely supported probability measure $\mu$ on $\mathrm{SL}_N(\mathbb{R})$ such that $\Gamma=\langle\operatorname{supp}\mu\rangle$ is discrete and Zar…

Let $\mathscr{V}^\lambda$ be the hyperbolic Poisson–Voronoi graph. For $r>0$, consider a geodesic in $\mathscr{V}^\lambda$ connecting the Voronoi cells containing $-r$ and $r$, and…

Let $K$ be a framed knot or link in $S^3$. Let $V_N(K)$ be the colored Jones polynomial corresponding to the $(N+1)$-dimensional irreducible representation of the quantum group…

Atiyah-Sutcliffe's determinant bound conjecture. For every $\mathbf{x}\in C_n(H^3)$,

Thurston's virtual fibration conjecture. Every finite-volume hyperbolic $3$-manifold has a finite cover that is fibered.

For a unit vector $cvec{a}=(a^1,a^2,a^3)$ and $s oindent\text{in}cmathbb{R}$, let … be the totally umbilic planes in the upper-half-space model $(cmathbb{R}^3_+,b)$ of hyperbolic…

Low-conformal-volume isotopy conjecture. If

Let $V_{\mathbb{A}_{1,k}}$ denote the volume of the moduli space of $(1,k)$-annuli, viewed as a polynomial in $\pi^2$. Here $k$ is a positive integer. Leading-term conjecture for a…

Let $\mathcal M_g$ denote the moduli space of closed hyperbolic surfaces of genus $g$, and let $\operatorname{sys}(S)$ be the systole length of $S$. Schmutz Schaller's systole-leng…

Let $K$ be an alternating hyperbolic link in $S^3$. Its hyperbolic volume is denoted by $\operatorname{vol}(K)=\operatorname{vol}(S^3\setminus K)$, and its determinant by $\det(K)$…

Let $M$ be a hyperbolic $3$-manifold with toroidal boundary and let $X$ be an ideal triangulation of $M$. Suppose the space of angle structures $\mathscr A_X$ is non-empty. Casson…

Generalized Andreev conjecture. There exists a compact convex hyperbolic polyhedron $Q$ combinatorially equivalent to $P$ whose dihedral angles are given by $\Theta$, and $Q$ is un…

A hyperbolic strictly-convex polyhedron is a strictly-convex polyhedron in hyperbolic space, with dihedral angles measured along its edges. Stoker's conjecture. Every hyperbolic st…

The volume conjecture for polyhedra. Then

Buser's conjecture. The surface $M$ has a pants decomposition in which each curve has length at most

Palindromicity conjecture. The Laurent polynomials $\det(\mathbf{A}(t))$ and $\det(\mathbf{B}(t))$ are palindromic.

Volume conjecture. As $r$ ranges over all odd integers and $q=e^{\frac{2\pi i}{r}}$,

Non-integral boundary slope conjecture. There does not exist an essential $n$-punctured sphere with non-meridional, non-integral boundary slope in a hyperbolic link exterior in the…

Alexandrov-Fenchel inequality. One should have

Let $\mathcal{M}$ be a $1$-cusped hyperbolic $3$-manifold. Let $\mathcal{M}(p/q)$ and $\mathcal{M}(p'/q')$ be its $p/q$- and $p'/q'$-Dehn-filled manifolds, respectively, and suppos…

Let $K$ be a hyperbolic alternating knot. Here, $\operatorname{vol}(K)$ denotes the hyperbolic volume of the complement $S^3\backslash K$, and $\det(K)$ denotes the determinant of…