99 problems

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…

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. The interior of a compact manifold with boundary is a topological model for a tame hyperbolic $3$-ma…

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 $M$.

Virtual Fibration Conjecture. $M$ has a finite cover which is a surface bundle over $S^1$.

Let $Y$ be a hyperbolic rational homology $3$-sphere, and let $H^{1,-}_0(Y\times\mathbb R)$ denote the relevant space of bounded solutions on the cylindrical anti-self-dual space…

Let $M$ be the hyperbolic $3$-manifold under consideration, let $B$ be its relevant deformation space, and let $\mathcal C$ be a non-empty collection of curves in $M$. Set … Let…

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 $K\subset{\mathbb S}^3$ 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 $K$ be a hyperbolic knot in ${\mathbb S}^3$, and let $l$ and $m$ denote the preferred longitude and meridian, respectively. A semi-geometric triangulation is an ideal triangula…

Let $K$ be a hyperbolic knot in ${\mathbb S}^3$, let $X$ be an ordered ideal triangulation of ${\mathbb S}^3{\smallsetminus} K$, and suppose that $X$ admits an angle structure. Let…

Let $L_1,L_2\subset T^2\times I$ be swatches such that the corresponding links $H\cup f(L_1)\subset S^3$ and $H\cup f(L_2)\subset S^3$ are three-component hyperbolic links. Let…

Let $L\thinspace\subset T^2\times I$ be an $m\times n$ swatch, and let $H$ be the Hopf link used to form the corresponding link $H\cup f(L)\subset S^3$. A swatch is reducible in te…