55 problems

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 finitely generated group with $g$ generators. For a hyperbolic $3$-manifold $N$, write $\operatorname{inj}_N(x)$ for the injectivity radius at $x$, and let the convex…

Let $\Gamma$ be a finitely generated, torsion-free, non-elementary Kleinian group, and let $\{\rho_n\}$ be a sequence in $\mathcal D(\Gamma)$ converging to a representation $\rho$.…

Let $G$ be a finitely generated group. Let $\operatorname{AH}(G)$ denote the space of algebraic limits of discrete faithful representations and let $\operatorname{MP}(G)$ denote th…

Let $\Gamma$ be a Kleinian group in $\operatorname{Mob}(\mathbb S^n)$ without parabolic elements, let $cd(\Gamma)$ denote its cohomological dimension, and let $\delta(\Gamma)$ deno…

Let $\Gamma$ be a finitely generated subgroup of $\operatorname{Mob}(\mathbb S^n)$ such that its limit set $\Lambda(\Gamma)$ has Hausdorff dimension less than $2$. Small-limit-set…

Let $\Gamma$ be a Kleinian group with limit set $\Lambda(\Gamma)$ of Hausdorff dimension less than $1$. The classical Schottky conjecture. The group $\Gamma$ is a Schottky-type gro…

Let $\Gamma\subset \operatorname{Mob}(\mathbb S^n)$ be an arithmetic lattice with $n\ge 3$. Kapovich–Potyagailo–Vinberg's conjecture. The group $\Gamma$ is noncoherent: it contains…

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

Let $M$ be a complete hyperbolic $3$-manifold with finitely generated fundamental group. A hyperbolic $3$-manifold is geometrically finite if its associated Kleinian group is geome…

Bers density conjecture. If $M$ is singly degenerate, then $\Gamma\in\overline{B}_X$, where $X$ is the conformal boundary of $M$.

Let $M$ be the fixed manifold parametrizing the deformation space ${\mathcal D}(M)$. A sequence $\rho_i\to\rho$ is type-preserving if parabolic elements are preserved throughout th…

Let $\Gamma$ be a finitely generated Kleinian group, with limit set $\Lambda$ in the Riemann sphere. Ergodicity conjecture. If … then $\Gamma$ acts ergodically on … This is describ…

Let $N$ be a complete hyperbolic 3-manifold with finitely generated fundamental group. An algebraic limit of geometrically finite hyperbolic 3-manifolds is a manifold obtained as t…

Let $M$ be the hyperbolic $3$-manifold under consideration, let $B$ be its relevant deformation space, and let $\mathcal C$ be a non-empty collection of curves in $M$. Set … Let…

Let $M$ be a complex curve equipped with two families of cuts satisfying the conditions in the preceding conjecture. For each integer $n\geq0$, let the associated set of $C^n$-poin…

Let $M$ be a complex curve. Let $\gamma_{i}^{\left(1\right)}$, $i\in I^{\left(1\right)}$, be a disjoint family of smooth embedded cycles in $M$ such that…

Let a Kleinian group be called admissible if it has generators $\varphi_i$, $i\in I_+$, satisfying the conditions of the amplification. For an admissible Kleinian group, a fundamen…

Let $G$ be a cocompact Kleinian group acting on the sphere at infinity $S^2_\infty$, and let $K\subset S^2_\infty$ be a compact connected set disjoint from every $G$-translate of i…

Let $G$ be a geometrically finite Bers boundary group, and let $\Lambda(G)$ be its limit set. Form the contact tree whose vertices are the components of the domain of discontinuity…

Let $K_i$, for $i\in\{1,2\}$, be Basilica limit sets of geometrically finite conformal dynamical systems. A map is weakly type-preserving if it sends parabolic points to parabolic…

Let $K_i$, for $i\in\{1,2\}$, be Basilica limit sets of geometrically finite conformal dynamical systems, such as limit sets of geometrically finite Kleinian groups or Julia sets o…

Let $\mathcal J$ be the Julia set of a rational map and let $\Lambda$ be the limit set of a Kleinian group. Suppose that $\mathcal J$ and $\Lambda$ are connected and neither is hom…

Let $\mathbb H^3$ be the upper half-space model of hyperbolic $3$-space, and let $\Gamma$ be a geometrically finite subgroup of…