131 problems

Penner's conjecture. Every pseudo-Anosov mapping class has a positive power obtained from Penner's construction.

Let $S$ be an orientable surface, let $M$ be a 3-manifold, and let $f\colon S\to M$ be a map. The induced homomorphism is … Simple Loop Conjecture. If $f_*$ is not injective, then…

Let $S$ be a surface, and let an object be naturally associated to $S$ and have sufficiently rich structure. Ivanov's metaconjecture. Every such object has the extended mapping cla…

Let $S$ be a finite-type orientable surface with finitely many marked points, or vertices, on its boundary and in its interior, and let $\mathcal{A}(S)$ be its arc complex. The pur…

Let $F$ be a closed surface, let $x_i\in F\times\{0\}$ for each $i$, and let ${\cal S}_{2,\infty}(F\times I,\{x_i\}_1^{2n};R,A)$ denote the relative Kauffman bracket skein module w…

Bianchi–Miller–Wilson's conjecture. For $k$ as above, the group $J^k\operatorname{Mod}(U)$ is exactly the kernel of the action of $\operatorname{Mod}(U)$ on…

Let $\Sigma_g$ be a closed surface, and consider the Torelli subgroup of its mapping class group, generated in particular by Dehn twists around null-homologous simple closed curves…

Let $\Sigma$ be an orientable surface with finitely many punctures and no boundary, and let $\mathcal{P}(\Sigma)$ denote its deformation space of complex projective structures. Eac…

Generic uniqueness conjecture. When $r$ belongs to a Zariski dense open subset of $\mathcal R_{\mathrm{SL}_2(\mathbb C)}(S)$, the representation $\rho_r$ is the only irreducible re…

Let $S$ be a finite-type hyperbolic surface, and let $\mathcal{E} \mathcal{L} (S)$ denote its ending lamination space with the coarse Hausdorff topology. Storm's connectivity conje…

Let $S$ be a connected oriented surface, let $\hat{\pi}=[S^1,S]$ be the set of free homotopy classes of oriented loops on $S$, and let $\mathbb{Q}\hat{\pi}$ be the Goldman Lie alge…

Let $S_g$ be a closed orientable surface of genus $g>2$, and let $M(S_g)$ denote its mapping class group. A subgroup of $M(S_g)$ is normal all pseudo-Anosov if it is normal and eve…

For a non-orientable surface, consider the values achievable by the relevant invariant discussed in the surrounding work. Massey's conjecture. The number of achievable values is li…

Luo's bounded-polytope conjecture. When $\lambda<0$, the image of $T(S)$ under the $\psi_{\lambda}$ edge-invariant coordinate is a bounded convex polytope.

Hougardy–Lutz–Zelke conjecture. Every triangulation of an orientable surface of genus $g$ with

Let $S_{g,p}$ be a genus-$g$ surface with $p$ punctures, let $\brun(S_{g,p})$ denote the set of Brunnian pseudo-Anosov mapping classes on $S_{g,p}$, and let $L(\brun(S_{g,p}))$ den…

Let $S$ be a surface, let $\mathcal L(S)$ denote its space of geodesic laminations, and let $d_{\log}$ be the metric on $\mathcal L(S)$ described in the source. Write $\delta$ for…

Stable torsion length conjecture. If

Positive stable torsion length conjecture. The stable torsion length of $t_a$ is positive:

An arc complex is the simplicial complex whose simplices encode collections of compatible arcs on a surface with boundary. Sphericity conjecture. The full arc complex of a surface…

Let $\Sigma_g$ be a closed surface of genus $g$, and let $\tau$ be a triangulation of $\Sigma_g$ whose lift to the universal cover is covered by a simple graph. Let…

Rokhlin's conjecture. Both Arnold surfaces ${\mathfrak A}_+$ and ${\mathfrak A}_-$ are standard.

Let $S$ be a surface, and let $\operatorname{Convex}(S)$ denote the space of convex curve functionals on $S$, equipped with the product topology inherited from its values on the cu…

Let $S$ be a surface with cusps and geodesic boundary. Let $\mathcal{C}_{\mathrm{cusp}}(S)$ be the class of curves and arcs in the cusped setting, and let…

Non-orientable surface complexity conjecture. Both $\mathsf{Pin}(\Sigma,s)$ and $\mathsf{SimplePin}(\Sigma,s)$ are in $\textsf{P}$ when $s\leq 3$, and are $\textsf{NP}$-complete wh…