Mathematics Problem Archive

If $M$ is a closed hyperbolic $3$-manifold, is every projective structure on $M$ convex?...

Let $N$ be the closed $3$-manifold obtained by gluing two copies of the figure-eight knot complement along their torus boundaries by a homeomorphism. ...

Let $N$ be a closed $3$-manifold whose JSJ decomposition contains only hyperbolic pieces. Does $N$ admit a convex projective structure?...

Geometric transitions are continuous paths of geometries that abruptly change type in the limit. Understand all transitions between the eight Thurston...

Does there exist a nonzero polynomial on $U\otimes V\otimes W$ invariant under $SL(U)\times SL(V)\times SL(W)$ when $\dim U=\dim V=4$ and $\dim W=8$?...

Let $\beta$ be a nondegenerate bilinear form on a finite-dimensional real vector space $V$, and let $G=\operatorname{Isom}(\beta)\subset GL(V)$. Which...

Let $M$ be a $3$-manifold whose fundamental group acts on a simplicial tree without global fixed points. In the covering space associated to an edge s...

Does every closed hyperbolic $3$-manifold contain a closed $\pi_1$-injective surface with only double curves of intersection?...

Does every closed hyperbolic $3$-manifold contain a closed $\pi_1$-injective surface with the 1-line property: in the universal cover, every pair of p...

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 $k$?...

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?...