35 problems

Let a lattice in $SU(1,n)$ act on the boundary of complex hyperbolic space. Boundary-action rigidity question. Is the action of any lattice in $SU(1,n)$ on the boundary of complex…

Let $\rho$ be an arithmetic isometric action of a lattice in $SU(1,n)$. Arithmetic-action deformation conjecture. Any sufficiently small perturbation of $\rho$ is of the form obtai…

Schwartz's conjectural picture. A $(p_1,p_2,p_3)$-representation is a discrete embedding if and only if neither $w_A$ nor $w_B$ is elliptic. The corresponding parameter values form…

Let $I$ be the critical interval for the family $\rho_t(p,q,r)$, let $\Gamma$ be the abstract $(p,q,r)$ triangle group, and for an infinite word $W\in\Gamma$ write $W_t=\rho_t(W)$.…

Let $(p,q,r)$ be a triple with $p\leq q\leq r$, let $\rho_t(p,q,r)$ be the associated family, and classify the triple as type A or type B according to the critical-interval definit…

Let $(p,q,r)$ be a triple with $p\leq q\leq r$, and let $\rho_t(p,q,r)$ be the associated one-parameter family. Call $(p,q,r)$ type A when the endpoints of the critical interval oc…

Let $\rho_t(p,q,r)$ be the one-parameter family of representations of the $(p,q,r)$-reflection triangle group by complex reflections, with $p\leq q\leq r$ and $\rho_0$ stabilizing…

Let $M$ be a compact complex surface of general type, and let $\operatorname{Vol}_{K}(M)$ and $\operatorname{Vol}_{s}(M)$ denote its minimal volume defined using sectional curvatur…

Schwartz's conjecture. The $(p,q,r;\alpha)$ representation of $\Delta_{p,q,r}$ is discrete and faithful if and only if $W_A$ and $W_B$ are non-elliptic. Furthermore, if $p<10$, it…

Let $f: \operatorname{\mathbb{B}}^m \rightarrow \operatorname{\mathbb{B}}^M$ be a proper holomorphic map that extends to a Hölder-continuous map…

Let $\mathbb{H}^m_{\mathbb{C}}$ and $\mathbb{H}^M_{\mathbb{C}}$ be complex hyperbolic spaces, and let $\Gamma < \mathsf{Isom}(\mathbb{H}^m_{\mathbb{C}})$ be a uniform lattice. Let…

Let $f: \operatorname{\mathbb{B}}^m \rightarrow \operatorname{\mathbb{B}}^M$ be a proper holomorphic map, and let $\operatorname{\mathsf{G}}_f$ denote its symmetry group. Suppose t…

Let $n\geq 1$ and let $\Gamma$ be a discrete and torsion-free subgroup of $\mathrm{PU}(n,1)$ acting on the complex hyperbolic space $\mathbb{H}^{n}_{\mathbb{C}}$. Define the critic…

Let $M$ be a compact complex hyperbolic quotient, and let $\mu$ be a semiclassical measure, meaning the semiclassical limit of a sequence of Laplacian eigenfunctions on $M$. Its su…

Dehn-surgery conjecture. The manifold at infinity of the even subgroup of $\Delta_{3,4,\infty;n}$ is the hyperbolic $3$-manifold obtained by Dehn surgery on the first cusp of…

Let $m$ be a positive integer, let $\mathbf{H}_{\mathbb{C}}^{m}$ be complex hyperbolic $m$-space, and let $\Gamma$ be a convex cocompact isometry group of…

Let $G_{6,3}=\langle a_0,\cdots,a_5\mid a_i^3=id,\ a_ia_{i+1}=a_{i+1}a_i,\ i\in\mathbb{Z}/6\mathbb{Z}\rangle$ be the hyperbolic group with boundary the Menger curve, and let…

Let $\Gamma<\operatorname{Aut}(\mathbf B^n)$ be a discrete, torsion-free subgroup, and let $M_\Gamma=\mathbf B^n/\Gamma$. For a positive integer $k$, suppose that…

Let $\Gamma<\operatorname{Aut}(\mathbf B^n)$ be a convex-cocompact, torsion-free subgroup, and let $M_\Gamma=\mathbf B^n/\Gamma$. Suppose that $\delta(\Gamma)=2$. The non-Stein cha…

There exists an integer $N$ such that for every $n\ge N$ and every lattice $\Gamma<\operatorname{PU}(n,1)$, the following hold: $\Gamma$ is not a reflection subgroup, and the under…

For every $n\ge 2$, consider arithmetic lattices $\Gamma_1,\Gamma_2<\operatorname{SU}(n-1,1)$ and $\Gamma_3<\operatorname{SU}(n-2,1)$. Suppose $\Gamma_3$ is isomorphic to subgroups…

Let $n$ be a positive integer. Nonarithmetic lattice existence conjecture. For each $n$, the group $\operatorname{PU}(n,1)$ contains a nonarithmetic lattice. Nonarithmetic lattices…

Let $G=\operatorname{Aut}(\mathbf B^n)$, and let $\Gamma<G$ be a convex-cocompact torsion-free subgroup. Let $\delta_\Gamma$ denote its critical exponent. Chengbo Yue's gap conject…

Let $\Gamma<\operatorname{PU}(n,1)$ be a discrete, torsion-free subgroup, let $\delta_\Gamma$ be its critical exponent, and let $k$ denote the dimension of a compact complex subvar…

The short-word discreteness conjecture. A $(n_1,n_2,n_3)$-complex hyperbolic triangle group is a discrete embedding in $PU(2,1)$ if neither $W_A$ nor $W_B$ is non-elliptic.