Unsolved Problems

(a) What is the maximum number of systoles on a closed, hyperbolic surface of genus $g$? (b) What is the maximum cardinality of a set of pairwise non-...

Let $S$ be a surface, and let $\Gamma_{1}, \Gamma_{2}$ be isotopy classes of embedded graphs in $S$. Determine when $\Gamma_{1}$ and $\Gamma_{2}$ are ...

(a) Does every Jordan curve in the Euclidean plane contain the vertices of a square? (b) Does every Jordan curve in the Euclidean plane contain the ve...

Is the genus $g$ Goeritz group $\mathcal{G}_{g}$ finitely generated when $g \geq 4$? If so, find a set of generators....

If two hyperbolic surfaces have the same unmarked simple length spectra (i.e., the same multiset of lengths that correspond to simple closed curves), ...

Suppose that $c$ is a geodesic current on a hyperbolic surface, and suppose that, on the space of hyperbolic metrics on the surface, $c$ has the same ...

What is the best lower bound on the volume of a fibered hyper- bolic 3-manifold one can give in terms of the translation length of the monodromy with ...

Which mapping classes give rise to arithmetic hyperbolic 3- manifolds as their mapping tori?...

Can one detect holomorphicity from a monodromy factoriza- tion of a Lefschetz pencil, fibration, or a surface bundle over a surface? What are the spec...

What is the minimum number, $m_{g,b}$, of right-handed Dehn twists along essential curves into which the boundary multi-twist, $\Delta:= T_{\delta_{1}...

Does every Lefschetz fibration over the 2–sphere admit a sec- tion?...

Does there exist a surface bundle over a surface that admits a complete hyperbolic metric, or, more generally, a complete metric of variable negative ...

Does there exist a complex surface $X$ that admits three or more non-isomorphic structures as a surface bundle over a surface?...

Consider surface bundles over surfaces where both fiber $F$ and base $B$ have genus $\geq 2$ and where $\pi_{1}(B)$ injects in the mapping class group...

(Kontsevich–Zorich conjecture). Understand the homotopy types of strata of abelian differentials. Which stratum-components are $K(\pi, 1)$ spaces? Wha...

For $n \geq 4$, does the braid group $B_{n}$ admit a finite-index sub- group that embeds in a right-angled Artin group?...

Let $\Gamma$ be a graph that is not a nontrivial join, and let $A(\Gamma)$ be the associated right-angled Artin group. Does there exist an injective m...

Determine the Artin groups that can be embedded into a map- ping class group....

Which right-angled Artin groups contain closed hyperbolic sur- face groups? Is there an algorithmic or graph-theoretic criterion to decide this?...

Let $S$ be a closed surface of genus at least 2. Show that the stable commutator length is rational on the commutator subgroup of $\pi_{1}(S)$....

Does every surface bundle over a surface admit a flat connec- tion? What about surface bundles over 3-manifolds?...

Let $S$ be a closed compact surface (without boundary). Is there a finitely generated, torsion-free group $G$ such that $G$ cannot act faithfully by h...

Let $S$ be a compact surface. For $0 \leq r < s$, does there exist a nontrivial finitely generated subgroup $G_{r} \leq \operatorname{Diff}^{r}_{0}(S)...

Is the first-order theory of the mapping class group of a surface decidable?...

Are systems of equations over mapping class groups and braid groups decidable?...

Give a Nielsen–Thurston-type classification for the mapping class groups of infinite-type surfaces. In particular, which homeomorphisms are the approp...

Give an appropriate analogue of the curve graph for infinite- type surfaces, and characterize the surfaces for which no such graph exists....

(a) Does the mapping class group of an infinite-genus surface with no planar ends contain every countable group? (b) Does the mapping class group of t...

(a) Let $S$ be an infinite-type surface and $\varphi$ a mapping class for which there is a (marked) conformal structure $\Sigma$ on $S$ with respect t...

Give a finite list of practically computable invariants of the mapping class group or pure mapping class group of an infinite-type surface $S$ that de...

Is the geodesic flow in almost every direction on the Chamanara surface ergodic? What about on the translation surface considered by Bruin and Lukina,...

Let $X$ be a compact, totally disconnected subset of $\mathbb{R}^{2}$ with $|X| \geq 2$, and let $\Gamma_{X}$ denote the mapping class group of $\math...

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?...