15 problems

Bers–Sullivan–Thurston Density Conjecture. Every hyperbolic $3$-manifold with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic $3$-mani…

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…

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…

Consider the arrowed graph $d4a4'$ in Figure SB2, which has two Hamiltonian cycles in distinct orbits under $\operatorname{Aut}(\Gamma')$, yielding two distinct degeneration diagra…

Let $d4a4_d4a4(K)$ and $d4a4_{d4a4'}(K)$ be the deformation spaces associated to the arrowed graphs $d4a4$ and $d4a4'$, and let $N(d4a4\xhookrightarrow{}d4a4')$ denote the relevant…

Let $S$ be the set of indices at which the projection of the Kisin variety to the corresponding factor of $(\mathbb P^1_{k_E})^f$ is constant, and write $0=i_1<i_2<\cdots<i_r$ for…

Let $D^\psi({\bf v},{\rm t},\overline\rho)$ denote the potentially Barsotti–Tate deformation space, and let $\overline{\mathscr{G\!R}}^\psi({\bf v},{\rm t},\overline\rho)$ be the a…

Finite-components conjecture.

Non-manifold conjecture.

Non-local-connectivity conjecture.

Bromberg's surface-group conjecture.

Bromberg's local-connectivity conjecture. Failure of local connectivity is a fairly widespread phenomenon in deformation spaces of hyperbolic $3$-manifolds.

Let $S$ be a surface, and let a Bers' slice for $S$ be the corresponding quasifuchsian deformation slice. Bers' slice conjecture. If $S$ is not a punctured torus or a $4$-times pun…

Let $(N,\mathcal P)$ be a pared $3$-manifold, and suppose that it has an essential cylinder, meaning an essential embedded annulus in the pared structure. Let $AH(N,\mathcal P)$ de…

Let $S$ be a compact surface with boundary. Write $AH(S\times[0,1],\partial S\times[0,1])$ for the deformation space of marked hyperbolic structures on the pared $3$-manifold…