26 problems

Let $\mathbb F_r$ be the free group on $r$ generators. A tuple $(w_1,\ldots,w_d)\in\mathbb F_r^d$ is simultaneously asymptotically equidistributed when its word-value distribution…

Let $S_n$ be the symmetric group and let $\mathbb F_r$ be the free group on $r$ generators. For a word with constants $w\in S_n\ast\mathbb F_r$, let $\mu_w^{S_n}$ be the distributi…

Let $F_n=\langle x_1,\ldots,x_n\rangle$ be the free group on $n$ generators. For a word $w\in F_n$, let $\operatorname{Pr}_w(g)$ be the probability that $w(g_1,\ldots,g_n)=g$ when…

Measure-equivalence conjecture. If $u$ and $w$ are measure equivalent in all finite groups, then there exists $\phi\in\operatorname{Aut}(F)$ such that

Hanany–Puder conjecture. For any $k\in\mathbb{Z}_{\geq2}$, any $\lambda\vdash k$, and any $w\in F_m$,

Let $F_r$ be the free group on $r$ generators. For $w,w'\in F_r$ and a compact group $G$, let $\mu_{w,G}$ be the push-forward of Haar measure on $G^r$ under the word map determined…

Word-map nonconstancy conjecture. Every non-constant word defines a non-constant word map.

Let $w(x_1,\dots,x_k)$ be a word in $k$ variables, and let $G$ be a profinite group. Write $w\{G\}$ for the set of word-values of $w$ in $G$, and $w(G)=\langle w\{G\}\rangle$ for t…

Let $w(x_1,\dots,x_k)$ be a word in $k$ variables, and let $G$ be a profinite group. Write $w\{G\}$ for the set of word-values of $w$ in $G$, and $w(G)=\langle w\{G\}\rangle$ for t…

Amit–Ashurst conjecture. Let $G$ be a finite nilpotent group. Then

Let $G$ be a finite nilpotent group, let $w$ be a word in $k$ variables, and define $N_w(g)$ to be the number of tuples $(g_1,\ldots,g_k)\in G^{(k)}$ satisfying…

Dominance conjecture. Then $\varphi_{w}$ is dominant. This extends the stated Lie-algebra analogue of Borel's theorem beyond the explicitly assumed $\mathfrak{sl}_{2}$ condition; t…

Let $K$ be an algebraically closed field of characteristic $0$, let $w(x_1,\dots,x_m)$ be a group word whose degree in each variable $x_i$ is $0$, and set…

Shalev's conjecture. For every finite simple non-abelian group $G$ and every $n\geq 1$, the $n$-th Engel word map

Let $G$ be a finite nilpotent group, let $\overline{x}$ denote a tuple of variables, and let $w(\overline{x})$ be a word map on $G$. Write $\mu_{G,w}(1)$ for the probability that…

Algorithmic radical-boundedness conjecture. There exists an algorithm which, on input $w$, decides whether there is a function

Effective Larsen–Shalev conjecture. The constants $N(w)$ and $\eta(w)$ can be computed algorithmically from the word $w$ as input.

Larsen–Shalev conjecture. For each such word $w$, there exist constants $N(w),\eta(w)>0$ such that, for every nonabelian finite simple group $S$ with $|S|\geq N(w)$,

Degree-zero word reformulation. The image of $w$ on $G$ must be

Let $K$ be an algebraically closed field of characteristic $0$, let $w(x_1,\dots,x_m)$ be a nontrivial group word, and let $\operatorname{PSL}_2(K)$ denote the projective linear gr…

Surjectivity conjecture. For every non-identity word $w\in F(x,y)$, the word map

For a fixed prime $p$, let $G_q$ be a family of Chevalley groups of fixed Lie type over $\mathbb{F}_q$, where $q=p^n$ varies. For a fixed word $w\in F_d$, with $d\ge 2$, let … be t…

Non-power word surjectivity conjecture. If $w$ is not a power word, then, for $G=\operatorname{PSL}(2,q)$,

Exceptional-value conjecture. Unless either $a=1$ and $\sqrt{2}\notin\mathbb{F}_q$, or the triple $(q,a,n)$ occurs in one of the explicitly listed exceptional cases, the map…

Engel-word surjectivity conjecture. For every $n\in\mathbb{N}$, the map $e_n$ is surjective for every finite simple non-abelian group $G$.