199 problems

For every genus $g$ and every finitely generated subgroup $G\leq \mathrm{Mod}(S_g)$, where $S_g$ is a closed oriented surface, the finite-index Frattini subgroup…

Let $T$ be the once-punctured torus, let $\mathcal{C}(T)$ denote the set of closed curves on $T$, and let $\mathcal{O}$ be any mapping class group orbit in $\mathcal{C}(T)$. For ev…

Let $n\geq 3$ be odd, and let $\Sigma_Q$ be the homotopy sphere obtained as the boundary of the manifold $Q$ in the relevant bordism group. Its class $[\Sigma_Q]$ lies in…

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…

Let $S_g^n$ be a compact oriented surface of genus $g$ with $n$ boundary components, and let $\mathrm{Mod}(S_g^n)$ be its mapping class group. Let…

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.

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

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…

Let $X=\mathbb{C}P^2\# n\overline{\mathbb{C}P^2}$ carry a symplectic form $\omega$, and let $\Gamma(X,\omega)$ be its Lagrangian root system. Write $PR[\Gamma(X,\omega)]$ for the p…

Let $S$ be a surface, let $\operatorname{Mod}(S)$ be its mapping class group, and let $\Lambda_G$ denote the limit set of a subgroup $G\leq \operatorname{Mod}(S)$ in the projective…

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 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 surface and let $G<\operatorname{Mod}(S)$ be parabolically geometrically finite, meaning that $G$ is relatively hyperbolic relative to a possibly trivial collection of…

Perron's conjecture. The map $\gamma_p$ is a well-defined invariant on $\mathcal{S}^3[p]$. Perron proposed this as an extension of the Casson invariant to the mod-$p$ Torelli group…

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…

Magic-manifold minimal-dilatation conjecture.

Let $\Sigma$ be a closed oriented surface of genus $g$, let $s^i\Sigma$ denote its $i$-th symmetric product, and let $HF_g^*$ be the Floer cohomology ring associated with…

Let $X$ be the $Z_n$ curve associated to the family in the source, with parameters $p$ and $q$, and let $g$ be its genus. A divisor of degree $g$ is non-special when it is not spec…

Let ${\cal M}_g$ be the mapping class group. Recall that a group ${\varGamma}$ is uniformly perfect if there is a natural number $N$ such that every element of ${\varGamma}$ is a p…

Let ${\cal M}_g$ be the mapping class group and let $H_b^2({\cal M}_g)$ denote its second bounded cohomology. Bounded-cohomology conjecture. For every genus $g$, the comparison map…

Let ${\cal K}_g$ be the Johnson kernel, let $d_i\in H^{4i-3}({\cal K}_g;{\mathbb Q})$ be the higher secondary classes, and let ${\cal M}_g$ act on $H^*({\cal K}_g;{\mathbb Q})$. Se…

Let ${\mathfrak h}^{\mathbb Q}_{g,1}$ be the rational graded Lie algebra of symplectic derivations, let $H_{\mathbb Q}$ be the rational symplectic module, and let…

Let $\operatorname{Map}_{g,1}$ be the mapping class group of a genus-$g$ surface with one marked point, and let $\operatorname{Map}_{g,1}(-)\subset\operatorname{Map}_{g,1}$ be the…

Let $C$ be a genus-$g$ Heegaard surface with fundamental group $\pi_1(C)$, let $X$ be the associated three-manifold, and let $L_+$ be one of the two lattices in the Heegaard splitt…