204 problems

Let $M$ be a finite-volume hyperbolic 3-manifold, and let $TV_r(M;e^{\pi i/r})$ be its Turaev–Viro invariant for odd integers $r\geq 3$. The Chen–Yang volume conjecture. The growth…

Let $S^{3}\setminus K$ be the complement of a hyperbolic knot $K$ in $S^{3}$. A closed embedded totally geodesic surface is a closed totally geodesic surface embedded in this hyper…

Let $M$ be a hyperbolic knot complement in $\mathbb{S}^3$. Two finite-volume hyperbolic 3-manifolds are commensurable when they share a common finite-degree cover; the commensurabi…

Let $K\subset\mathbb{S}^3$ be a hyperbolic knot, and let $\mathbb{S}^3\setminus K$ denote its complement. The complement has hidden symmetries when a homeomorphism between finite-d…

Lubotzky–Sarnak conjecture. For some sequence of finite-index normal subgroups intersecting only at the identity, the Cayley graphs of the corresponding finite quotients do not for…

Weeks manifold minimal-volume conjecture. The Weeks manifold attains the minimum volume among hyperbolic $3$-manifolds.

Let $K$ be a hyperbolic knot in ${\mathbb S}^3$. An ideal triangulation $X$ of ${\mathbb S}^3{\smallsetminus} K$ has a nonempty angle-structure space $\mathcal{A}_X$, and let…

Let $K\subset{\mathbb S}^3$ be a hyperbolic knot, let $\gamma$ be a simple closed curve on $\partial\nu(K)$, and let $\rho_0$ be the discrete faithful representation associated wit…

Let $M$ be a hyperbolic $3$-manifold, either closed, cusped or with totally geodesic boundary, and let $\operatorname{Vol}(M)$ denote its volume. As $r$ varies over the odd integer…

Let $\varphi$ be a pseudo-Anosov homeomorphism of a punctured surface, and let $M_\varphi$ be its hyperbolic mapping torus. The Bonahon–Wong–Yang invariants are defined at roots of…

Let $F \subset \mathbb{C}$ be a number field not contained in $\mathbb{R}$, and let $\mathcal{M}$ be the set of manifolds with invariant trace fields contained in $F$, … Let…

Let $M$ be a closed, orientable, arithmetic, hyperbolic $3$-manifold. The Short Geodesic Conjecture. There is a constant $C>0$, independent of $M$, such that the length of the shor…

Let $M$ be a compact $3$-manifold, and let $AH(M)$ denote the relevant space of marked hyperbolic structures introduced in the source. Bers–Sullivan–Thurston density conjecture. Th…

A closed hyperbolic 3-manifold is a closed 3-manifold admitting a complete hyperbolic metric, and its fundamental group is denoted by $\pi_1(M)$. A finite index subgroup…

A finitely generated Kleinian group is a discrete subgroup of the group of Möbius transformations with finite generation as an abstract group. A quasi-conformal deformation of a ge…

Kahn–Marković's counting conjecture. There exists a constant $C(M)$ such that

A flow on a $3$-manifold is quasigeodesic if each flowline is coarsely comparable to a geodesic, and it is pseudo-Anosov if it has a transverse contracting-expanding structure gove…

Let $M$ be a hyperbolic knot complement in $\mathbb{S}^3$ with hidden symmetries. A cusp of an orbifold is a $(2,3,6)$-cusp when its cross-section is the Euclidean triangle orbifol…

Profinite volume conjecture. The profinite completion of the fundamental group of a finite-volume hyperbolic $3$-manifold determines its hyperbolic volume.

Let $G<\operatorname{Homeo}(S^2)$ be a geometrically finite convergence group. Martin–Skora's covering conjecture. Then $G$ is covered by a Kleinian group. Here, being covered mean…

Let $(M,g)$ be a closed Riemannian $3$-manifold, let $R_g$ denote its scalar curvature, and let $h(g)$ denote its volume entropy. Agol–Storm–Thurston conjecture. If … then … This c…

Let $(N_3,\gamma)$ be the acylindrical taut sutured manifold obtained by cutting the 3-chain link complement along a 3-punctured sphere. Its volume is $5.33\ldots$. Minimal-volume…

Let $M_f$ be a hyperbolic $3$-manifold with Heegaard surface $\Sigma$ of minimal genus, and let ${\mathcal D}_1$ and ${\mathcal D}_2$ be the disk sets of the two handlebodies. For…

A link in the $3$-sphere is a knot if it has one component, and a link complement has a right-angled structure when its relevant geometric structure is right-angled. Champanerkar–K…

Let $G$ be an arithmetic hyperbolic $3$-manifold group, and consider a decreasing sequence of congruence subgroups of $G$ with trivial intersection. The first homology torsion grow…