767 problems

Let $M$ be a closed hyperbolic $3$-manifold. Thurston's virtual fibering conjecture. The manifold $M$ has a finite cover that fibers over the circle, equivalently, that is the mapp…

Thurston's bending conjecture. The data of these two pleating loci determine the geometry of the entire manifold $M$.

Let $K\subset S^3$ be a hyperbolic knot, let $g(K)$ denote its genus, and let $\mathcal T_K(t)\in\mathbb C[t^{\pm1}]$ be the normalized hyperbolic torsion polynomial associated wit…

In the space $X^n$, let $d_n(r)$, for $n\geq 2$, denote the density of $n+1$ mutually tangent spheres, or horospheres when $r=\infty$, of radius $r$ relative to the simplex spanned…

Let $K$ be a hyperbolic knot, meaning that its complement admits the hyperbolic Thurston geometry, and let $J_r(K;q)$ denote its $r$-th colored Jones polynomial. Kashaev–Murakami–M…

A hyperbolic $3$-manifold is algebraically finite when its fundamental group is finitely generated, and it is topologically finite when it is homeomorphic to the interior of a comp…

Schmutz Schaller's conjecture. The multiplicity of the simple length spectrum is bounded above by $6$:

Ehrenpreis conjecture. There are such finite covers for which the Teichmüller distance between $M_\epsilon$ and $N_\epsilon$ in the suitable moduli space is less than $\epsilon$.

The paper studies stretch maps between hyperbolic surfaces and the associated canonical geodesic laminations. Higher-dimensional extension conjecture. There should be an extension…

Let $N$ be a compact $3$-manifold with boundary, and let $\mathcal{T}_N$ be an ideal triangulation of $N$. A truncated hyperbolic tetrahedron realization assigns hyperbolic truncat…

Every compact hyperbolic $3$-manifold with totally geodesic boundary is a compact $3$-manifold equipped with a complete hyperbolic metric whose boundary is totally geodesic. A geom…

Let $K$ be a hyperbolic knot, let $J_r(K;q)$ be its $r$-th colored Jones polynomial, and let $A_*$ denote the geometric flat connection associated with the canonical hyperbolic str…

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 $F_2$ be the rank-two free group, and consider a type-preserving representation $\rho:F_2\to\mathrm{PSL}(2,\mathbb{R})$, meaning a non-elementary representation for which the p…

Let $S$ be a closed orientable surface of genus $k\geq 2$. For a quasi-Fuchsian hyperbolic 3-manifold, let its convex core be the minimal compact totally convex subset, whose bound…

Let $M$ be a hyperbolic $3$-manifold with toroidal boundary, let $X$ be an ideal triangulation of $M$, and let $\gamma\in\pi_1(\partial M)$ be a simple closed curve. For…

Let $\Gamma$ be a uniform lattice in $SL(2,\mathbb{R})=\operatorname{Isom}(\mathbb{H}^2)$, so that $\mathbb{H}^2/\Gamma$ is a closed orbifold, and let a random walk on $\Gamma$ hav…

Geometric triangulation conjecture. Every cusped hyperbolic $3$-manifold admits a geometric triangulation.

Gromov–Lawson–Thurston conjecture. A necessary condition for the disc bundle to admit a hyperbolic structure is

Eigenvalue lower-bound conjecture. All eigenvalues satisfy

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

Ros's conjecture. Any open orientable surface can be properly embedded in ${\mathbb H}^2\times {\mathbb R}$ as a minimal surface.

For a closed oriented connected $n$-dimensional manifold $M$ admitting a hyperbolic metric, let $g$ be a Riemannian metric on $M$, let $\|M\|_{\Delta}$ denote its Gromov simplicial…

A closed hyperbolic four-manifold is a closed four-dimensional manifold equipped with a hyperbolic structure. Its Seiberg--Witten invariants are the invariants associated with its…

Martin boundary conjecture. Almost surely, $\partial \mathbb{D}$ is a realization of the Martin boundary of $G$.