99 problems

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 ${\cal M}$ be a closed hyperbolic $3$-manifold. A surface subgroup is the image of the fundamental group of a closed surface of genus at least $2$ under a map inducing an injec…

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, 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 with a regular triangulation $\mathcal{T}$ and NeumannZagier datum, including a choice of quad type with $\det \mathbf{B}\neq 0$. Let…

Let $M$ be a hyperbolic $3$-manifold with finitely generated fundamental group. The interior of a compact manifold with boundary is a topological model for a tame hyperbolic $3$-ma…

Chen–Yang Turaev–Viro volume conjecture.

Let $F$ be a free group, and let $H,K\le F$ be subgroups with equal rank $m\ge 2$. If … then Guzman's group-theoretic conjecture. … Here $H\vee K$ is the subgroup generated by $H$…

Let $M$ be a fixed closed hyperbolic $3$-manifold, and let $M_n$ be covers of $M$ that Benjamini–Schramm converge to hyperbolic $3$-space $\mathbb{H}^3$. The analytic torsion conve…

A hyperbolic Mom-$4$ manifold is a hyperbolic $3$-manifold equipped with a Mom-$4$ structure. Mom-4 conjecture. The collection of hyperbolic Mom-$4$ manifolds is exactly the set of…

Subexponential regulator conjecture.

Let $M$ be a closed hyperbolic $3$-manifold, and let $b_1(M)$ denote its first Betti number. Volume lower-bound conjecture. If … then … This conjectural lower bound is used to esta…

Let $N$ be a hyperbolic 3-manifold and let $c_1$ and $c_2$ be closed geodesics in $N$. Their complements are the manifolds obtained by removing the geodesics. Closed geodesic compl…

Let $N$ be a cusped hyperbolic 3-manifold, and consider fillings of $N$ along distinct slopes that produce hyperbolic manifolds. A filling pair is cosmetic if the resulting manifol…

Fibered-cover characterization conjecture. The fundamental group $\pi_1(N)$ contains a subgroup which is locally free but not free if and only if $N$ does not have a finite cover w…

Universal locally free subgroup conjecture. The fundamental group $\pi_1(N)$ contains a subgroup which is locally free but not free.

Let $G$ be a finitely generated group with $g$ generators. For a hyperbolic $3$-manifold $N$, write $\operatorname{inj}_N(x)$ for the injectivity radius at $x$, and let the convex…

Let $\Gamma$ be a finitely generated, torsion-free, non-elementary Kleinian group, and let $\{\rho_n\}$ be a sequence in $\mathcal D(\Gamma)$ converging to a representation $\rho$.…

Let $G$ be a finitely generated group. Let $\operatorname{AH}(G)$ denote the space of algebraic limits of discrete faithful representations and let $\operatorname{MP}(G)$ denote th…

Let $M$ be a hyperbolic $3$-manifold with invariant trace field $k=k(M)$, and let $\overline{k}$ denote its complex conjugate. The Chern–Simons invariant $CS(M)$ lies in…

Let $M$ be a hyperbolic $3$-manifold with finitely generated fundamental group. Its topological type consists of its underlying topology, and its end invariants encode the geometry…

Let $M$ be a cusped manifold, and let $(W_n,L_n,\rho_n)$ be a sequence of compact hyperbolic Dehn fillings of $M$ converging geometrically to $M$, where $L_n$ consists of the simpl…

Let $M$ be a cusped manifold, define … and let $H_N(M)$ denote the proposed quantum hyperbolic invariant. Here ${\rm CS}(M)$ and ${\rm Vol}(M)$ are respectively the metric Chern–Si…

Bers density conjecture. If $M$ is singly degenerate, then $\Gamma\in\overline{B}_X$, where $X$ is the conformal boundary of $M$.