23 problems

Distance-five conjecture.Delta(alpha,beta) leq 5.

Let $M$ be a complete finite-volume hyperbolic $3$-manifold, and consider the space of its Dehn filling deformations. Hodgson–Kerckhoff conjecture. This space is connected and smoo…

Let $M$ be a hyperbolic 3-manifold with one cusp, and let $N$ and $N'$ be Dehn fillings of $M$. Matveev–Fomenko conjecture. If $c(N)<c(N')$, then ${\rm Vol}(N)<{\rm Vol}(N')$. The…

Let $N_{p/q,r/s,t/u}$ be the closed Dehn filling of $N=M6^3_1$ with coefficients $\\{p/q,r/s,t/u\\}$. Completeness conjecture for filling repetitions. The repetitions among filling…

Let $S$ be any surface of finite type, and let $g_1,\dots,g_l\in \mathcal{MCG}^\pm(S)$. For a subgroup $H$ of a group $G$, write $\langle\langle H\rangle\rangle$ for its normal clo…

Let $M$ be a hyperbolic once-punctured torus bundle, let $\widetilde{C}$ denote the relevant character-curve component, and let $\mathcal{A}_{k(\widetilde{C})}$ be its tautological…

Let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold. Let $\mathcal{M}_{p_1/q_1, \dots, p_n/q_n}$ and $\mathcal{M}_{p'_1/q'_2,\dots, p'_n/q'_n}$ be two Dehn fillings of…

Let $M$ be a one-cusped, finite-volume, hyperbolic $3$-manifold, and let $M_{m/n}$ be a hyperbolic Dehn filling with surgery coefficient $m/n$. Write $\Gamma=\pi_1(M_{m/n})$, and l…

Let $M$ be a tunnel-number-one orientable $3$-manifold with incompressible torus boundary. A tunnel of $M$ determines a distinguished-wave-determined slope on $\partial M$. The two…

Let $\mathcal{M}$ be a $1$-cusped hyperbolic $3$-manifold with non-quadratic cusp shapes. For $n>0$, choose $n$ distinct Dehn filling coefficients, all sufficiently large, and let…

Let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold with non-symmetric cusp shapes. Let $N_{\mathcal{M}}^{\mathbb{C}}(v)$ denote the number of Dehn fillings of $\mathcal{M}$…

Let $K$ be a hyperbolic knot in $S^2\times S^1$, and let a Dehn surgery on $K$ mean the corresponding Dehn filling of its exterior. Cabling conjecture for $S^2\times S^1$. No hyper…

Let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold. Uniform boundedness conjecture. There exists a constant $c=c(\mathcal{M})$ such that, for every volume $v$, the number o…

Let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold, and let $\mathcal{M}_{(p_1/q_1,\dots,p_n/q_n)}$ denote the corresponding hyperbolic Dehn filling. Finite coordinatewise…

Let $\mathcal{M}$ be a $1$-cusped hyperbolic $3$-manifold, and write $\mathcal{M}_{p/q}$ for its hyperbolic Dehn filling along slope $p/q$. Finite linear parametrization conjecture…

Let $\mathcal{M}$ be a $1$-cusped hyperbolic $3$-manifold with volume function $\Theta_{\mathcal{M}}(x,y)$, and let $\mathcal{Z}$ be the analytic variety defined by … Let…

Planar-surface horoball-packing conjecture. For any horoball packing of $F$, there is at least one horoball of area less than

Meridian length conjecture. Four is the maximal length of a meridian of a knot complement.

Link-volume slope finiteness conjecture. For any $V>0$, there exists a finite set of slopes $\mathcal{F}$ on $\partial X$, so that if $\lv[X(\alpha)] < V$, then $\alpha$ intersects…

Let $M$ be a finite-volume hyperbolic $n$-manifold with torus cusps, let $V$ be its complete hyperbolic interior, and let $M(T_1,\ldots,T_m)$ denote the manifold obtained by fillin…

A non-hyperbolic Dehn filling is a Dehn filling of a $1$-cusped hyperbolic $3$-manifold that does not remain hyperbolic. Strong Gordon conjecture. The figure-$8$ knot complement is…

Finiteness conjecture. There are only finitely many one-cusped hyperbolic $3$-manifolds with more than six exceptional Dehn fillings.

Classification conjecture. There are only four orientable hyperbolic manifolds with two exceptional Dehn fillings $r_1,r_2$ satisfying