Mathematics Problem Archive

Do cusped finite-volume hyperbolic $3$-manifolds have closed quasi-Fuchsian surfaces that cubulate except for the cusps? Equivalently in the stated ge...

Are finite-volume hyperbolic $3$-manifolds virtually semi-fibered?...

Which Kleinian groups admit closed quasi-Fuchsian surface subgroups?...

Let $M$ be a $3$-manifold, let $\phi:\pi_1M\to\mathbb{Z}$ be dual to $(\Sigma,\partial\Sigma)\subset(M,\partial M)$, and let $N$ be obtained by cuttin...

Is there a practical algorithm to test whether a pair of $3$-manifolds are homeomorphic?...

The CAT(0) cubical dimension of a group $G$ is the least dimension of a CAT(0) cubical space on which $G$ acts geometrically. Is there a sequence of c...

Do closed hyperbolic $3$-manifold groups have finite-index subgroups that embed in a word-hyperbolic reflection group?...

Does every freely indecomposable cubulated hyperbolic group contain the fundamental group of a closed hyperbolic surface?...

If $M$ is a closed $3$-manifold, when does $\pi_1M$ act on the affine building for $SL(4,\mathbb{R})$ so that the quotient retracts to $M$?...

Let $\Gamma$ be a discrete group acting properly discontinuously by affine transformations on $\mathbb{R}^5$. Is $\Gamma$ virtually an extension of a ...

Classify all properly discontinuous affine actions of a given closed surface group on $\mathbb{R}^6$....

For a given right-angled Coxeter group, what is the least $n$ for which it admits a proper affine action on $\mathbb{R}^n$?...

Characterize hyperbolic $3$-manifolds with infinitely generated fundamental group. In particular, is there a $3$-manifold that is locally hyperbolic, ...

Do there exist fibered hyperbolic $3$-manifolds that are homology $S^2\times S^1$ and have arbitrarily large injectivity radius? Are there hyperbolic ...

Characterize the Thurston norm polytopes of finite-volume hyperbolic $3$-manifolds....

Does every hyperbolic $3$-manifold have a finite-sheeted cover homeomorphic to a CAT(0) cube complex?...

Fix $\mu$ below the three-dimensional Margulis constant. For hyperbolic $3$-manifolds of volume less than $V$, drill all closed geodesics of length le...

If $M_1$ and $M_2$ are cusped hyperbolic $3$-manifolds, does there exist a cover $M'_1\to M_1$ and a nonzero-degree map $M'_1\to M_2$ taking cusps to ...

The renormalized volume of quasi-Fuchsian groups gives a function $\rho:\mathcal{T}(S)\times\mathcal{T}(S)\to\mathbb{R}$. Is $\rho$ a metric on the Te...

Give a rigorous explanation for why SnapPy works so well in practice....

Given $R>0$ and an integer $n\geq4$, is there an $\varepsilon>0$ and a finite set of hyperbolic $n$-simplices such that every closed cone $n$-manifold...

Build a combinatorial model for hyperbolic $3$-manifolds with explicit bilipschitz constants....

Let $\Gamma$ be a Kleinian group of finite covolume, and let $\mathcal{C}(\Gamma)$ be the set of isomorphism classes of its finite quotient groups. Do...

For $\Gamma=F_r$ with $r\geq2$, does $\mathcal{C}(F_r)$ determine $F_r$ up to isomorphism?...

Let $\Gamma$ be a Kleinian group of finite covolume. Does its set $\mathcal{C}(\Gamma)$ of finite quotient isomorphism classes determine $\Gamma$ amon...

Let $M$ be a hyperbolic $3$-manifold and let $n$ be any positive integer. Does $M$ admit an ideal triangulation with $m\geq n$ positively oriented tet...

If $G$ is a Gromov-hyperbolic group whose boundary is homeomorphic to the limit set of a convex-cocompact Kleinian group, is $G$ virtually a convex-co...

Which subsets of $S^2$ can occur as limit sets of convex-cocompact Kleinian groups?...

For which Kleinian groups does the limit set contain a Sierpiński carpet? For which Kleinian groups does the limit set contain a continuum?...

Characterize the limit sets of graph Kleinian groups and iterated graph-Kleinian groups. A graph Kleinian group is a convex-cocompact Kleinian group f...

Show that a random walk on the mapping class group gives a pseudo-Anosov element whose invariant foliations have generic trivalent singularities with ...

Start with $n$ tetrahedra and glue their faces together at random. The vertex links need not be spheres but are essentially random triangulated surfac...

Rivin's experimental results appear extremely regular, possibly indicating additional structure. Investigate this phenomenon....

For the orbit of a point $x$ in Teichmüller space under the mapping class group, show that as $r\to\infty$: (1) the proportion of orbit points in the ...

Let $f(x)\in\mathbb{Z}[x]$ be irreducible over $\mathbb{Q}$ with all roots real, let $f(\alpha)=0$, let $k=\mathbb{Q}(\alpha)$, and let $\mathcal{O}_k...

For $\mathcal{O}_k$ as in item 7.1, let $\Gamma_{\mathrm{unit}}$ have vertex set $\mathcal{O}_k$, joining two elements when their difference is a unit...

If $k$ is a number field that is not totally real, is there a hyperbolic $3$-manifold with trace field $k$?...

Can there be a degenerate Kleinian group that is not the fiber of a fibration and has algebraic trace field?...

Is there a singly degenerate Kleinian group for which all matrix entries of all group elements lie in one fixed number field?...

Let $M$ be a finite-volume hyperbolic $3$-manifold. If $M$ contains infinitely many immersed totally geodesic surfaces, must $M$ be arithmetic?...

Do Haken hyperbolic $3$-manifolds have strongly irreducible Heegaard splittings?...

For a hyperbolic $3$-manifold with torus boundary, does its profinite completion determine whether it is a knot complement?...

Is there an algorithm to detect whether a Heegaard splitting is reducible and, if so, find a reducing curve?...

Is there a classification of the strongly irreducible Heegaard splittings of a given $3$-manifold?...

Do higher-dimensional smooth manifolds always admit multisections? What is the correct generalization of uniqueness up to stabilization for multisecti...

A band joining the components of a two-component link $L\subset S^3$ is called complicated if either its core cannot be isotoped to meet a splitting s...

Study the function Ψ: Mg→ [−log(µg),∞) as a (rational) Morse function. Classify its rational critical points. Deduce properties of the rational cohomo...

Determine the finiteness properties of Ig. For which k is Hk(Ig) finitely gen- erated? For which k is there a K(Ig, 1) with finite k-skeleton (one say...

Letγ1,···,γ 2g be the standard basis of Z2g. Study the function L = ∑ Lγi:Yg→ [0,∞) as a Morse function. Find critical sets and deduce properties of t...

Work out the details of this construction of the completion Yg of Yg. Show that L:Yg→ [0,∞) extends to L:Yg→ [0,∞) and is a proper map. Ideally, inclu...