15 problems

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 $g\geq 1$, let $\mathcal{D}_g$ be the Dehn quandle of a closed orientable surface of genus $g$, and let $\mathbb{Z}[\mathcal{D}_g]$ be its integral quandle algebra. An element…

Let $M$ be a pinched negatively curved manifold of infinite volume, of dimension $n$, and let $[hat{\omega}]in H^{n}_b(M,\mathbb{R})$ denote its bounded fundamental class. Kim–Kim'…

Monod's isomorphism conjecture. The continuous bounded cohomology of $G$ with real coefficients is naturally isomorphic to its ordinary continuous cohomology:

Bowden's conjecture. The stable mapping class group $\Gamma_\infty$ is boundedly acyclic.

Let $M$ be a complete Riemannian manifold of dimension $n$ with a properly discontinuous, cocompact, isometric action by a group $G$, and let … be the isomorphism obtained from the…

Let $X$ be a closed Riemannian manifold, and let $\alpha \in H^2(X;\mathbb{R})$. The class $\alpha$ is $\widetilde{d}$-bounded if its pullback to the universal cover…

Let $M$ be a closed oriented manifold covered by $(\mathbb{H}^2)^n$, and let $\Theta_n$ be the bounded volume form defined in the preceding conjecture. Simplicial-volume equality c…

Let $G$) be a connected, non-compact, semi-simple Lie group with finite center. Write … for the comparison map from continuous bounded cohomology to continuous cohomology. Dupont's…

Let $F_2$ be the free group of rank $2$, let $\operatorname{PSL}_2\mathbb{C}$ be the group of orientation-preserving isometries of hyperbolic $3$-space, and let…

Let $G$ be a connected semisimple Lie group with finite center. Write $H^\bullet_{\mathrm{cb}}(G;\mathbb{R})$ for continuous bounded cohomology and…

Let $G$ be a connected noncompact simple Lie group and $X$ an irreducible symmetric space. Given a geodesic $\gamma$ in $X$, let $F(\gamma)$ be the union of all geodesics in $X$ pa…

Stable bounded-cohomology conjecture. The bounded cohomology of $\Gamma_{\infty}$ is trivial:

Higher-degree comparison-map conjecture. The comparison map