25 problems

Masur–Minsky's conjecture. The diameter of the set of vertex cycles for every train track in every surface is bounded above by $3$.

Schleimer's conjecture. Every “well-behaved” multiarc and curve graph is equipped with a witness-based distance estimate formula.

Let $G$ be an Artin-Tits group of spherical type, let $\mathcal C(G)$ be its curve graph, and let $\alpha\in G$. Suppose a vertex of $\mathcal C(G)$ has a finite orbit under…

Let $G$ be an Artin-Tits group of spherical type with rank $n$, let $\mathcal C(G)$ be the curve graph, and let $\alpha\in G$. Consider a vertex of $\mathcal C(G)$ with a finite or…

Let $\Sigma$ be a compact, connected, orientable surface, and let $\mathcal A$ be a multiarc and curve graph on $\Sigma$. For a witness $W$, let…

Let $\Sigma$ be a compact, connected, orientable surface, and let $\mathcal A$ be a multiarc and curve graph on $\Sigma$. A compact, essential subsurface $W\subset\Sigma$ is a witn…

Linear higher connectivity conjecture. Gabai's higher connectivity results for Gromov boundaries of curve graphs should strengthen to linear higher connectivity.

Let $a_0$ and $a_4$ be curves on the closed surface $S_g$ with $d(a_0,a_4)=4$, and let $T_{a_4}$ denote the Dehn twist about $a_4$. Two curves are minimally intersecting when their…

Let $\Sigma$ be a connected, compact surface of genus $g=1$, and let $\mathcal{C}^\dagger(\Sigma)$ denote the appropriate fine curve graph, whose vertices are smooth essential clos…

Finite-invariance index one conjecture. If

Let $B_n$ be the $n$-strand braid group, and let $\mathcal C_{\rm AL}(B_n)$ be its additional length graph. Let the curve graph of the $(n+1)$-punctured sphere be the graph whose v…

Let $H$ be a handlebody with $m\geq 3$ spots. The disk graph of $H$ has vertices given by isotopy classes of essential disks, with edges joining classes admitting disjoint represen…

Ivanov's conjecture. The automorphism group of a natural complex associated to a surface should be the extended mapping class group, and the proof should factor through the automor…

Let $\alpha$ and $\beta$ be the endpoints of a shortest origami edge-path, and let $\alpha_i$ and $\alpha_{i+1}$ be consecutive curves on it, with $\alpha_{i+1}$ intersecting…

Intersection-quotient conjecture. Under this condition, the edge-path is a quasi-geodesic.

Let $S$ be the surface under consideration, and let $\mathcal{SC}(S)$ denote the Schmutz Schaller curve graph, with vertices and edges as defined in the paper. Let…

Let $A_n$ be the braid group with its standard Artin generators, let $\mathcal{C}_{AL}(A_n)$ be its additional length graph, and let $\mathcal{D}_n$ be the associated punctured dis…

Relative hyperbolicity classification conjecture. $d4a2$ is relatively hyperbolic if and only if this HHS structure has isolated orthogonality.

Let $S$ be a surface and let $v$ and $w$ be curves in its curve graph, with spiral surgery understood as the surgery operation described in the paper. Generalized spiral-surgery co…

Let $S$ be a surface, let $(v,w)$ be a filling pair of curves, and let ${\mathfrak{G}}$ be the collection of efficient geodesics between $v$ and $w$. Let $\mathbb{S}$ denote a surg…

Optimality conjecture. For $r>1$, this is also the optimal asymptotic upper bound. In particular, there should be examples attaining the corresponding lower order $g^{-1-1/r}$ alon…

The paper has established rigidity and automorphism results for certain mapping class groups of infinite-type surfaces. Further-applications conjecture. Further applications of the…

Finite-invariance index one conjecture. If

Let $S$ be a surface and let $\tau$ be a train track on $S$. A vertex cycle of $\tau$ is the simple closed curve whose counting measure represents an extreme point of the projectiv…

Let $S$ be a surface with complexity $\xi(S)$, let $k$ be an integer, and let $\alpha$ and $\beta$ be simple closed curves on $S$. Write $d_{\mathcal{C}}$ for curve graph distance…