72 problems

Let $\rho$ be a complex hyperbolic ultra-parallel $[0,0,0]$-triangle group representation, with generators $R_1,R_2,R_3\in\mathrm{PU}(2,1)$. Goldman–Parker conjecture. Such a repre…

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…

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…

Schwartz's conjecture. The representation is discrete and faithful if and only if neither $W_A$ nor $W_B$ is elliptic.

Let $\Delta(p,q,r;\theta)\subset \operatorname{PU}(2,1)$ be the complex hyperbolic triangle group generated by complex reflections $I_1,I_2,I_3$, and let … be the corresponding abs…

Superrigidity conjecture. Maximal representations of a complex hyperbolic lattice into a Hermitian Lie group should be superrigid.

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…

Holomorphic section conjecture. There exists a holomorphic section of $M$ in all these examples.

Mostow's commensurability conjecture. In each of these nine cases, $\Gamma$ is commensurable with a group obtained from the theorem describing the discrete monodromy groups of fini…

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 group $\Delta(p,q,r)$ is discrete and faithful if and only if both $W_A$ and $W_B$ are non-elliptic.

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 $\Gamma$ be a convex-cocompact and torsion-free subgroup of $\mathrm{PU}(n,1)$ acting on the complex hyperbolic space $\mathbb{H}^{n}_{\mathbb{C}}$, and suppose that its critic…

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 manifold of complex dimension two equipped with a Kähler–Einstein metric and having negative sectional curvature. Si–Yau's conjecture. The manifold $M$…

Chain-link ancestry conjecture. The chain link $C_p$ is an ancestor of the 3-manifold at infinity of the critical complex hyperbolic triangle group $\Delta_{p,q,r;\infty}$, for…