123 problems

Let $(G,\mathcal{P})$ be a relatively hyperbolic group pair whose Bowditch boundary is homeomorphic to $S^2$. Relative Cannon conjecture. Then $G$ acts properly, isometrically, and…

Let $F_n$ be a free group and let $\rho\colon F_n\to\operatorname{PSL}_2(\mathbb{C})$ be a discrete, faithful representation without parabolic elements. Minsky's primitive-stabilit…

Let $\mathcal{F}_a$ be the correspondence in the family described above, $\mathcal{M}$ the Mandelbrot set, and $\mathcal{M}_{\Gamma}$ the modular Mandelbrot set, namely the set of…

Let $G$ be a finitely generated Kleinian group, and let $\Lambda(G)$ denote its limit set on the sphere. Ahlfors's conjecture. The limit set $\Lambda(G)$ is either the entire spher…

Let $(u,v)$ be a generator pair for $F_2$, and let $\rho\colon F_2\to SL(2,\mathbb C)$ be an irreducible representation satisfying the $BQ$-conditions: for every primitive element…

Let $M$ be a compact $3$-manifold, and let $AH(M)$ denote the relevant space of marked hyperbolic structures introduced in the source. Bers–Sullivan–Thurston density conjecture. Th…

Meromorphic-power conjecture. Then there is an integer $N$ such that

A finitely generated Kleinian group is a discrete subgroup of the group of Möbius transformations with finite generation as an abstract group. A quasi-conformal deformation of a ge…

Let $S$ be a closed surface, let $\operatorname{MCG}(S)$ act by precomposition on $\mathfrak{X}(\pi_1(S),\operatorname{PSL}_2(\mathbb{C}))$, and let $\mathfrak{X}_{QF}$ denote the…

Let $\Gamma\subset \mathsf{PSL}(d,\mathbb R)$ be a subgroup that is Borel Anosov, meaning that it is $P$-Anosov for a Borel subgroup $P$. Sambarino's conjecture. If $\Gamma$ is Bor…

Let $G<\operatorname{Homeo}(S^2)$ be a geometrically finite convergence group. Martin–Skora's covering conjecture. Then $G$ is covered by a Kleinian group. Here, being covered mean…

A Riemann surface is a surface equipped with a complex structure. A classical Schottky uniformization is a Schottky uniformization for which there exists a fundamental domain in…

Bullett–Freiberger general mating conjecture. All polynomials with connected Julia set can be mated with Hecke groups as correspondences.

Classification conjecture. The same classification is true for groups generated by two elements of finite order.

Let $\Gamma$ be a finitely generated discrete subgroup of $\operatorname{PSL}_2(\mathbb{C})$, and let $\Lambda$ be its limit set in $\mathbb S^2$. Ahlfors measure conjecture. The l…

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…

Let \rho:\pi_1(\text{\boldmathT})\to PSL(2,\mathbb{C}) be a type-preserving representation such that its end-invariant set satisfies $\mathcal{E}(\rho)=\Lambda(\hat\Gamma_r)$.…

Let $N$ be a hyperbolic $3$-manifold with finitely generated fundamental group. Ahlfors' Measure Conjecture. The limit set of $N$ has measure zero or is the entire Riemann sphere.…

Let $S$ be a surface and let $AH(S \times I)$ denote the deformation space of Kleinian groups associated to the product $S \times I$. Bromberg's conjecture. The space…

Let $G$ be a finitely generated Kleinian group, and let $AH(G)$ denote its deformation space of faithful discrete representations of $G$ into $PSL_2\mathbf{C}$, modulo conjugacy, t…

A Kleinian group is a discrete subgroup of the group of isometries of hyperbolic $3$-space, and its co-volume is the hyperbolic volume of the corresponding quotient orbifold. Let…

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 group that is not virtually abelian, let ${\cal D}(G)$ be the deformation space of representations of $G$, and let $\{\rho_j\}\subset{\cal D}(G)$ converge algebraicall…

Let a sequence of isomorphic Kleinian groups converge algebraically. Its algebraic limit is the limiting Kleinian group in the algebraic topology, and its geometric limit is the li…