40 problems

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 $M^n$ be a closed manifold. In part (a), assume that $M^n$ is a flat Riemannian manifold. In part (b), assume that $M^n$ admits an almost flat structure. Let $W^{n+1}$ be a com…

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

Metric-independence conjecture. For every such metric $g$, the delocalized $L^2$-invariants are well-defined and independent of the choice of $g$.

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

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

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…

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

Kobayashi's conjecture. The canonical bundle $K_X$ is ample.

Hyperbolic spectral exponential sum conjecture. As $X,T\to\infty$, for every $\epsilon>0$,

Quantum variance conjecture. One has

Farrell–Zdravkovska conjecture.

Let $M= obreakbackslash \mathbb{H}^n$ be a closed hyperbolic $n$-manifold with fundamental group $\Gamma$, where either $nleq 3$ or $M$ is arithmetic of simplest type. A **$\lambda$-q…

Let $\G$ be a subarithmetic hyperbolic group with growth exponent $\ga$, and let $\cO$ be the ring of integers of its trace field. The group $\G$ asymptotically length-saturates if…

Let $M$ be a finite-volume hyperbolic manifold of dimension at least $4$. Let $f\colon M\to S^1$ be a non-homotopically trivial smooth map such that $\ker(f_*)$ is finitely generat…

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…

Cubulation conjecture. Assume that either $n \leq 3$ or that $M$ is arithmetic of simplest type. Then, for every $\lambda > 1$, there is a compact cubical model $X$ such that there…

Q-curvature volume comparison conjecture. If

Let $d=2n+1$ and let $\Gamma\subset \operatorname{SO}_0(d,1)$ be an arithmetic subgroup. Let $\{\Gamma_i\}$ be a cusp-uniform sequence of torsion free congruence subgroups of…

Let $M$ be a closed hyperbolic manifold, let $g$ be an auxiliary Riemannian metric on $M$, and let $V_{hyp}(R)$ denote the volume of a geodesic ball of radius $R$ in hyperbolic spa…

Let $N$ be a compact Riemannian manifold with negative sectional curvature, let $(\Psi_n)_{n=0}^{\infty}$ be an orthonormal basis of Laplace eigenfunctions on $N$, and let $\mu_n$…