44 problems

Let $w(x_{1},x_{2})$ be a word in two variables, and let ${\operatorname{PSL}}_{2}({\mathbb{F}}_q)$ denote the projective special linear group over the finite field with $q$ elemen…

Let $F_r$ be a free group of rank $r$, let $w\in F_r$ be a word, and let $\lambda\vdash k$ be a partition. For sufficiently large $n$, write $\lambda^{+}(n)$ for the stable partiti…

Let $G$ be a finite nilpotent group, let $w$ be a word in $k$ variables, and let $N_w(1)$ denote the number of tuples $(g_1,\ldots,g_k)\in G^{(k)}$ such that $w(g_1,\ldots,g_k)=1$.…

Let $G$ be a connected compact simple real linear algebraic group, and let a word map be the map induced by a word in finitely many variables on $G$. Larsen's conjecture. Every wor…

Let $w_1$ and $w_2$ be non-trivial group words, and for a group $G$ let $w_i(G)$ denote the set of values of $w_i$ in $G$. Two-word Waring conjecture. There exists an integer…

Ashurst's conjecture. Every fiber of a word map on $G$ satisfies

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 its word-value…

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 $S_n$ be the symmetric group, let $\mu_w^{S_n}$ be the push-forward of normalized counting measure on $S_n^r$ under the word map induced by $w\in\mathbb F_r$, and let…

Let $S_n$ be the symmetric group on $n$ letters, let $mathbb Fr$ be the free group on $r$ generators, and let $winmathbb Fr$ induce the word map $wcolon Sn^rto Sn$. A word is a **…

Let $F_r$ be a free group of rank $r$, let $w\in F_r$, let $k$ be a positive integer, and let $\lambda\vdash k$. Write $\lambda^{+}(n)$ for the stable partition obtained by adjoini…

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

Amit's conjecture. The probability of evaluating to the identity satisfies

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.

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 $\underline{G}$ be a semisimple algebraic group with Lie algebra $\mathfrak{g}$. Let $w_{0}=xyx^{-1}y^{-1}$ be the group commutator word and let $w=[X,Y]$ be the Lie algebra co…

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…

Let $K$ be algebraically closed of characteristic $0$, let $w=w(x_1,\dots,x_m)$ be a nontrivial group word, and let $e_{12}$ be the standard elementary matrix. Unipotent-value conj…

Let $K$ be an algebraically closed field of characteristic $0$, and let $w(x_1,\dots,x_m)$ be a nontrivial group word. PSL2 word-map conjecture. The image of $w$ on…

Let $r\in\mathbb{Z}_{\ge 1}$, let $\mathbb{F}_r$ be the free group on generators $x_1,\ldots,x_r$, and let $w\in\mathbb{F}_r$. For a finite group $G$, the $w$-measure is the push-f…