403 problems

Let $\mathfrak{M}_{\infty,1}$ be the homotopy colimit of the genus-stabilisation maps … For each $i\geq 1$, let $\kappa_i$ denote a class of degree $2i$ in the rational cohomology…

Let $S=S_{g,r}^n$ be an orientable surface of genus $g\ge 3$ with $r$ punctures and $n$ boundary components, and let…

Let $S$ be a closed oriented surface, and let $\operatorname{Mod}(S)$ be the pure extended mapping class group acting on the character variety $\mathfrak{X}(S)$. The action preserv…

Let $\Sigma_{g,n}$ be a compact oriented surface, and let … be its mapping class group, with homeomorphisms and isotopies fixed on the boundary. For an odd integer $r\geq 3$, let…

For each $g\geq 3$, let the genus-$g$ Goeritz group of $S^3$ be the group of orientation-preserving homeomorphisms of the genus-$g$ Heegaard splitting of $S^3$, modulo isotopy. Pow…

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

Let $\Sigma$ be the surface under consideration, let $\widetilde{\Sigma}\to\Sigma$ be a finite characteristic cover, and let $v\in {\rm H}_1(\widetilde{\Sigma},\mathbb{Q})$ be a no…

Putman–Wieland conjecture. For every $g\ge 2$, every $p\ge 0$, and every finite-index subgroup $K<\pi_1(\Sigma^p_g,\star)$, the subgroup $K$ has the Putman–Wieland property.

Let $\mathcal{I}_g$ be the Torelli subgroup of the mapping class group of the genus-$g$ oriented surface, consisting of mapping classes acting trivially on…

Let $(\Sigma,[g])$ be a Riemann surface with marked points, let $\mathcal{M}_{\mathfrak{g},m}$ be its moduli space, and let $\mathcal{T}_{\mathfrak{g},m}$ be Teichmüller space, the…

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 $\Sigma_{g,n}$ be a surface of genus $g$ with $n\leq 1$ boundary components, and let $\mathcal{I}(\Sigma_{g,n})$ denote its Torelli group. Johnson's cubic-growth conjecture. Fo…

Three-involution generation conjecture. The upper bound of four in the stated generation results is not sharp: $\operatorname{Map}(S(n))$ is generated by three involutions for all…

Let $\Sigma$ be a surface, let $\operatorname{MCG}(\Sigma)$ denote its mapping class group, and let $S$ be a finite generating set for $\operatorname{MCG}(\Sigma)$. Write $d_S$ for…

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…

Even-odd monotonicity conjecture. For all $k\geq 1$,

Let $g$ be the genus, let $\mathfrak{t}_g$ be the relevant graded Lie algebra associated to the Torelli group, and let $\mathfrak{m}_g$ be the Lie algebra of symplectic derivations…

Let $(\Lambda,[I],K)$ be the triple corresponding to a pseudo-Anosov map $\phi\in Mod~(X)$, and let $\lambda_\phi$ be its dilatation. The integral order $\Lambda$ has class number…

For a character $[\rho]\in\mathcal{X}$, an end invariant is a lamination $\lambda\in\mathscr{PL}$ for which there exists a sequence of distinct $X_k\in\mathscr{C}$ converging to…

Let $S$ be a closed surface, let $\operatorname{MCG}(S)$ act by precomposition on $\mathfrak{X}(\pi_1(S),\operatorname{PSL}_2(\mathbb{C}))$, and let $\mathfrak{X}_{QF}$ denote the…

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…

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

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 $B_n$ be the Artin braid group and let $\Gamma_{g,k}$ be the mapping class group of an orientable surface $S_{g,k}$ of genus $g$ with $k$ boundary components. Let $B_{\infty}$…

Johnson's amalgamation conjecture. The group $\operatorname{Isot}(M,\Sigma)$ is an amalgamation of the two groups $\operatorname{Isot}(V^-_{\Sigma'},\Sigma^-)$ and…