55 problems

Volume rigidity conjecture. The metric $g$ is isometric to the hyperbolic metric $g_H$.

Farrell–Zdravkovska conjecture. Every flat manifold bounds geometrically; equivalently, every flat manifold occurs as the cusp cross-section of a one-cusped hyperbolic manifold.

SKT-to-Kähler hyperbolicity conjecture. If $X$ admits such a metric $\omega$, then $X$ is Kähler hyperbolic.

Let $\mathcal{M}_n$ be the class of closed oriented $n$-manifolds, and let $\gtrsim$ denote domination by a continuous map of nonzero degree. Let…

Let $G=SO(n,1)$ and let $\Gamma$ be an arithmetic lattice in $G$. Serre's conjecture. The group $\Gamma$ fails the congruence subgroup property. The notes say this conjecture was a…

Let $\Gamma$ be any lattice in $\mathop{\mathrm{SO}}\nolimits(1,n)$. Thurston's conjecture. There are a finite-index subgroup $\Gamma'$ of $\Gamma$ and a surjective homomorphism ……

A finite-volume hyperbolic manifold is a manifold equipped with a complete finite-volume hyperbolic metric; its fundamental group is cubulated if it acts properly and cocompactly o…

Cusp covering conjecture. For each cusp $C$ of $M$, there exists a sublattice $\Lambda_C$ of $\pi_1(C)$ such that, for every choice of a sublattice $\Lambda'_C\subseteq\Lambda_C$ f…

A hyperbolic lattice is a lattice in the isometry group of hyperbolic space, and a group is coherent if every finitely generated subgroup is finitely presented. Kapovich's conjectu…

Let $\Gamma\subset\operatorname{Isom}(\mathbb H^n)$ be a lattice, let $\Gamma'\subset\Gamma$ be a finite-index subgroup, and let $\rho$ denote the relevant natural representation.…

Let $\Gamma\subset \operatorname{Mob}(\mathbb S^n)$ be a lattice. Virtual first Betti number conjecture. There exists a finite-index subgroup $\Gamma'\subset\Gamma$ such that … Equ…

Let $M=\mathbf{R}^n/\Gamma$ be a closed flat $n$-manifold, where $\Gamma$ is a discrete, torsion-free, cocompact subgroup of $\operatorname{Isom}(\mathbf{R}^n)$. A cusp cross-secti…

Conjecture B. There exists a finite congruence covering $\widehat{M}$ of $M$ such that the lift of $F$ to $\widehat{M}$ has the asserted homological and cohomological realization p…

Conjecture A-minus. For every natural number $i\leq\frac{d_G}{2}-1$, there exists a constant $\varepsilon(G,i)>0$ such that, for every congruence subgroup $\Gamma$ of $G$,

Conjecture A. If $G^{\rm nc}\cong SO(2n,1)$, then, for every natural number $i\leq n$,

Stretching cusps conjecture. The sequence $\{(M,g_i)\}$ Gromov–Hausdorff converges, in the appropriate sense, to a disjoint union of two isometric copies of $N$.

Hamiltonian volume conjecture. The rescaled volume satisfies

Mohammadi–Oh's conjecture. The optimal condition $\delta_\Gamma>d/2$ suffices for $L^2(\Gamma\backslash G)$ to admit a strong spectral gap for the complementary series, in arbitrar…

Counting conjecture. There exists a constant $C(M)$ such that

Let $(N^n,g)$ be an orientable closed $n$-manifold with $\operatorname{Sc}_g\geq -n(n-1)$, and let $(M^n,g_H)$ be an orientable closed hyperbolic $n$-manifold. Suppose there exists…

Let $M$ be a closed, real-analytic Riemannian manifold with negative sectional curvature, and suppose that $M$ contains infinitely many closed totally geodesic immersed hypersurfac…

A finite-volume hyperbolic manifold is a quotient of real hyperbolic space by a group of isometries acting freely and properly discontinuously. RFRS hyperbolic virtual fibring conj…

Let $G$ be the fundamental group of a finite-volume hyperbolic manifold of odd dimension, assume that $G$ is RFRS, and let $\beta_i^{(2)}(G;\mathbb K)$ denote its $i$th $L^2$-Betti…

A finite-volume hyperbolic manifold is a quotient of real hyperbolic space by a group of isometries acting freely and properly discontinuously. Odd-dimensional hyperbolic virtual f…

Rigidity conjecture. If $Y$ is isomorphic to $\mathbb{P}^1$ or an elliptic curve, then $\mathcal{H}$ is empty or the whole $Y$.