10 problems

Commutator abelianness conjecture. The commutator subgroup

Let $B_n$ be the braid group on $n$ strands. For a braid, its closure is the link obtained by joining corresponding endpoints. A subgroup $H$ of $B_n$ is non-trivial normal if it i…

Let $\mathcal{M}$ be a countable recursively saturated model of $\mathrm{PA}$, let $\mathrm{G}:=\operatorname{Aut}(\mathcal{M})$, and let $I$ be a cut of $\mathcal{M}$. Define … Th…

Let $G$ be a finite group with a solvable subgroup $S$ of index $n$. The Babai–Goodman–Pyber polynomial-index conjecture says that the absolute constant $c$ in the bound for a solv…

Let $G$ be a finite group. Say that $G$ has distinct conjugacy-class sizes when any two distinct conjugacy classes of $G$ have different cardinalities. The $S_3$-conjecture. If a f…

Let $\Gamma$ be the arithmetic group specified by the local-to-global conjecture above, fix a non-principal ultrafilter $\mathcal{N}$ on the natural numbers, and let $O^*$ and…

Let $S_{g,n}$ be a genus-$g$ surface with $n$ punctures, let $\Mod(S_{g,n})$ be its mapping class group, and let $L_C(H)$ denote the least asymptotic translation length on the curv…

Let $S_g$ be a closed orientable surface and let $N$ be a normal subgroup of $\operatorname{Mod}(S_g)$ that is isomorphic to a right-angled Artin group. Normal right-angled Artin s…

Let $S_g$ be a closed orientable surface, let $N$ be a normal subgroup of $\operatorname{Mod}(S_g)$, and call a subsurface not large if it is contained in a nonseparating subsurfac…

Normalish subgroup conjecture. Every normalish subgroup of $T$ contains an embedded copy of R. Thompson's group $F$ or an embedded non-abelian free subgroup.