Unsolved Problems

Given an infinite-type surface $S$, which homeomorphisms $f: S \to$ $S$ give rise to mapping tori $M_{f}$ that admit a hyperbolic structure? For those...

Compute the end-periodic cobordism group $\Delta^{e}_{2}$ of end-periodic automorphisms (diffeomorphisms or homeomorphisms) of surfaces....

(a) Which coarsely boundedly generated mapping class groups of infinite-type surfaces are hyperbolic? (b) Consider the class of surfaces with $n \geq ...

(a) Given a mapping class $\psi$ of a based surface $S$, there is an induced endo- morphism of the symmetric product $\operatorname{Sym}^{i}(S)$ and h...

The mapping class group of a closed, orientable, genus $g$ sur- face $S$ acts by symplectomorphisms on the symmetric product $\operatorname{Sym}^{g}(S...

(AMU conjecture). Let $S$ be a surface with negative Euler char- acteristic. If $\varphi \in \operatorname{Mod}(S)$ acts by a pseudo-Anosov on some su...

(Volume conjecture for surface diffeomorphisms). Let $S$ be a closed oriented surface, let $q=e^{2\pi i/n}$ be a root of unity, and let $\mathcal{K}^{...

Classify the smallest volume hyperbolic 3-manifolds of various types. In particular: (a) Determine the nonorientable closed hyperbolic 3-manifolds of...

Show that the volumes of hyperbolic 3-manifolds are not all rationally related....

Does every cusped hyperbolic 3-manifold have a geometric ideal triangulation?...

(Chen--Yang Volume Conjecture). (a) Prove that, for any hyperbolic 3-manifold $M$, $$ \lim_{\substack{r\to\infty\\ r\ \mathrm{odd}}}\frac{1}{r}\log\b...

(a) Do there exist closed non-Haken hyperbolic 3-manifolds with arbitrarily large injectivity radius? (b) Does there exist a cofinal tower of regular...

Given a cofinal tower of covers M $\leftarrow$ $M_1$ $\leftarrow$ $M_2$ $\leftarrow$ $\cdots$, is it true that the torsion subgroups $\operatorname{To...

Does every finite-volume hyperbolic 3-manifold admit a finitesheeted cover fibering over the circle with orientable pseudo-Anosov monodromy?...

If $M_1$ and $M_2$ are finite-volume hyperbolic 3-manifolds whose fundamental groups have isomorphic profinite completions, must $M_1$ and $M_2$ be is...

Is being Haken a profinite invariant amongst 3-manifolds? That is, if $M_1$ and $M_2$ are 3-manifolds so that $\pi_1(M_{1})$ and $\pi_1(M_{2})$ have i...

(a) Are there infinitely many commensurability classes of arithmetic rational homology 3-spheres? (b) Are there infinitely many arithmetic integral h...

Does every hyperbolic knot in the 3-sphere have meridian length at most 4?...

(a) Considering all closed, orientable, $\pi_1$-injective surfaces (possibly non-embedded) in all closed hyperbolic 3-manifolds, what is the infimum o...

Does every closed hyperbolic 3-manifold admit an immersed $\pi_1$-injective surface with only double points? More precisely, if M is a closed, connect...

Can a hyperbolic knot complement in the 3-sphere contain a closed, embedded totally geodesic surface?...

Let $M$ be a closed hyperbolic 3-manifold with positive first Betti number. (a) Which elements of $H^{2}(M;\mathbb{R})$ are realized as the Euler cla...

Does every finite-volume hyperbolic 3-manifold contain infinitely many simple closed geodesics?...

Let $M_1$ and $M_2$ be finite-volume hyperbolic n--manifolds. If the length spectra of $M_1$ and $M_2$ coincide, must the two manifolds be commensurab...

Is there a closed hyperbolic 3-manifold that is foliated with minimal leaves?...

(a) Does every closed hyperbolic 3-manifold have a nowhere zero vector field whose lift to the universal cover has proper flow lines? (b) Can one ens...

Let $M$ be a closed hyperbolic 3-manifold with a faithful homomorphism $\rho:\pi_1(M)\to \operatorname{Homeo}^{+}(\mathbb{R})$. Prove that $M$ support...

In this problem, all 3-manifolds are orientable, while all flows are considered up to orbit equivalence and are assumed to be transitive. (a) Are the...

Let $G=\pi_1(M)$ be the fundamental group of a finite-volume hyperbolic 3-manifold $M$. What is the regularity of the smoothest (virtual) action of $G...

What is the Margulis constant in dimension 3? Is it realized uniquely by the Weeks manifold W, where $\mu(W)$ = 0.77442...?...

(a) (Cannon Conjecture) If G is a finitely presented, Gromov hyperbolic group with space at infinity equal to the 2-sphere, must G be a cocompact Klei...

(Bending Conjecture). (a) Is a quasi-Fuchsian group determined by the hyperbolic metric on the boundary of its convex core? (b) Is a quasi-Fuchsian g...

Let $M$ be a finite-volume hyperbolic 3-manifold, and let $M^1$ be a minimal-index finite cover of $M$ such that $\pi_1(M^1)$ embeds in a right-angled...

(a) What is the computational complexity of the homeomorphism problem for compact, orientable 3-manifolds? (b) Is there a polynomial-time algorithm t...

How many Pachner moves are needed to pass between two triangulations of a compact 3-manifold?...

(a) Given a closed hyperbolic 3-manifold $M$, can one find an explicit bound on the degree of a finite cover $\widetilde M$ having $b_1(\widetilde M)>...

(a) What is the computational complexity of determining whether a compact 3-manifold admits a hyperbolic structure? (b) If a compact 3-manifold does ...

Suppose M is a closed 3-manifold. (a) Can one decide if the fundamental group of M is left-orderable? (b) What is the complexity of a certificate of...

Is there an algorithm to determine whether two closed, embedded surfaces in $\mathbb{R}^3$ are isotopic?...

Let M and N be closed orientable 3-manifolds. Prove that if there is a degree-1 map $f:M\to N$ then $g(M)\geq g(N)$, where $g(M)$ is the Heegaard genu...

Do any two genus-g Heegaard splittings of a closed, orientable 3-manifold M become equivalent after at most g stabilizations?...

Given a compact manifold M, let $r(M)$ denote the rank of its fundamental group and $g(M)$ denote its Heegaard genus. (a) Does every closed orientabl...

(Simple loop conjecture) Let f : F $\to$ M be a 2-sided immersion of a surface into a 3-manifold such that $f_*$ : $\pi_1(F)$ $\to$ $\pi_1(M)$ is not ...

(a) Is every finitely generated 3-manifold group linear (over some field with characteristic zero)? (b) If so, can one bound the dimension of a faith...

Is every PD$_3$-group the fundamental group of a closed, aspherical 3-manifold?...

(a) Is every finitely generated perfect group the normal closure of a single element? (b) Is there an integral homology sphere whose fundamental grou...

Does every closed, orientable, hyperbolic 3-manifold admit a tight contact structure?...

Is it true that for every knot $K\subset S^3$, there is an integer $n_K$ such that $S^3_r(K)$ admits a tight contact structure for all $r\geq n_K$?...

Does every tight contact 3-manifold have finite Giroux torsion?...

Understand how various properties of contact structures behave under different kinds of symplectic cobordism. For instance: (a) Is tightness preserve...