28 problems

A train-track on the surface $S$ has vertices that determine simplices in the curve complex, and let $d_{\mathcal C}$ denote distance in the curve complex. The constant $B$ is a bo…

Let $X$ have a single boundary component $F$ with $g(F)\geq 2$, and let $V\cup_H W$ be a minimal-genus Heegaard splitting of $X$ with $F$ contained in the compression body $V$. Let…

Local-to-global rigidity conjecture. The pants complex exhibits local-to-global rigidity: local combinatorial data determine the corresponding global structure, in the sense that l…

Let $\Sigma$ be a connected surface and let $B_r(o)$ be the radius-$r$ ball in its curve complex, with $o$ an origin. A ball is $N$-almost simply connected if every loop in…

Let $\Sigma=\Sigma_{g,n}$ be a connected surface with genus $g$ and $n$ punctures, with complexity $\xi(\Sigma)=3g-3+n\geq 10$. Let $C\Sigma$ be its curve complex and let…

For a set of minima, let $C_{\mathrm{max}}$ be the largest set of curves containing $C$ for which every nonfilling subset $C'$ of $C_{\mathrm{max}}$ gives…

Let $\Gamma$ be a finite-index subgroup of $\operatorname{Mcg}(\Sigma)$, and let $\alpha$ be a simple closed curve on $\Sigma$. A subgroup of $\operatorname{Mcg}(\Sigma)$ is congru…

Let $L_{\C}(k,g)$ denote the minimal asymptotic translation length on the curve complex among pseudo-Anosov mapping classes of genus $g$ with homological dilatation parameter at le…

Let $\operatorname{Mod}(S)$ be the mapping class group of a surface $S$, let $\mathcal{C}(S)$ be its curve complex, and let $f\in\operatorname{Mod}(S)$ have asymptotic translation…

Let $S_{g,n}$ be a genus-$g$ surface with $n$ punctures, let $\Mod(S_{g,n})$ be its mapping class group, let $\PMod(S_{g,n})$ be the pure mapping class group, and let $L_{\C}(G)$ d…

Let $\gamma$ be a Teichmüller geodesic with endpoints $\gamma^-$ and $\gamma^+$ in the Thurston boundary, and call it sticky when it satisfies the sticky-geodesic condition that ge…

Let $S$ be a finite-type surface, let $\mathcal{S}\subset\mathcal{PMF}$ be the set of simple closed curves, let $d_\star$ be the star metric, and let $d_\mathcal{C}$ be the distanc…

Let $\mathcal{A}$ be a central hyperplane arrangement with associated bordification $X$ and boundary $\partial X$. Assume that $\mathcal{A}$ satisfies properties A, B, C, and D, an…

Let $\Sigma$ be a surface, let $\mathcal{C}_R(\Sigma)$ be a complex of regions, and let $N$ be a normal subgroup of $\operatorname{Map}^{\pm}(\Sigma)$. Brendle–Margalit conjectures…

Baik–Shin–Wu's sharpness conjecture. For each $d\geq 2$, there exist $M$, $\mathcal{F}$, and $\Omega$ such that $\mu_d^{\mathcal{C}}$ is bounded away from $0$ on an infinite subset…

Let $S_g$ be a closed orientable surface and let $\mathcal C_A(S_g)$ be a complex of regions associated to $S_g$. Assume that $\mathcal C_A(S_g)$ is connected and admits no exchang…

Let $S_g$ be a closed surface of genus $g$, and let $\mathcal C_A(S_g)$ be a complex of regions that is connected and has no holes or corks. Complex-of-regions conjecture. The natu…

Let $K$ be a knot in a 3-manifold $M$ with a $(g,b)$-splitting, where $g\geq 1$. Let $(g',b',n)$ be a triple of non-negative integers with $g'\geq g$ and $b'\geq b$. High-distance…

Electric-model quasi-isometry conjecture. The map $\varphi$ is a quasi-isometry.

Projected normal-form quasi-geodesic conjecture. This family of paths forms a uniform family of unparametrized quasi-geodesics in $\mathcal{CC}$.

Braid-group quasi-isometry conjecture. The additional length complex is quasi-isometric to the curve complex:

Let $S$ be a surface such that … and let $\{\alpha_{1}, \ldots, \alpha_{n}\} \subset \mathcal{C}(S)$ be a simplex. Optimization conjecture. There exists a simplicial superinjective…

Short Curve conjecture. If $g$ is a Weil–Petersson geodesic with end invariant $(\nu^{-},\nu^{+})$, then:

Let $S$ be a finite-type surface. Let $\mathcal{C} (S)_{\operatorname{sub}}$ be the subspace model of the curve complex, and let…

Let $S$ be a $p$-punctured surface of genus $g$, set $n=3g+p-4$, and let $\mathcal{E} \mathcal{L} (S)$ denote its ending lamination space. Dimension duality conjecture. The dimensi…