17 problems

Profinite rigidity conjecture for word automorphic orbits. If there exists $\widehat{f}\in\operatorname{Aut}(\widehat{F}_k)$ such that

Let $1\ne w\in\mathbb{\mathbf{F}}$, let $\mu=(\mu_{1},\ldots,\mu_{k})$ be a partition, and let $e_{w,\mu}$ be the number of stable-square-length-extremal surfaces, orientable or no…

Let $1\ne w\in\mathbb{\mathbf{F}}$, let $\mu=(\mu_{1},\ldots,\mu_{k})$ be an integer sequence with no zero entries, and let $d_{w,\mu}$ be the number of orientable stable-square-le…

Let $\mathbb{\mathbf{F}}$ be a free group. Let $1\ne w\in\mathbb{\mathbf{F}}$ be a non-power, let $\sigma\in S_{d}$, and suppose that … is an element of…

Let $\mathbb{\mathbf{F}}$ be a free group and let $w_{1},w_{2}\in\mathbb{\mathbf{F}}$. For each word, consider the induced probability measure on every compact group. Word-measure…

Let $\mathbb{\mathbf{F}}$ be the free group, let $w\in\mathbb{\mathbf{F}}$, and let $\chi$ be a non-trivial stable irreducible character of an arbitrary sequence of groups with sta…

Let $\mathbb{\mathbf{F}}$ be the free group, let $\mathrm{ssql}(w)$ denote the stable square length of a word $w$, and let ${\cal I}_{U}$, ${\cal I}_{O}$ and…

Let $G$ be a finite group, let $F_r$ be the free group of rank $r$, and let $w\in F_r$. For a stable irreducible character $\chi=(\chi_n)_{n\in\mathbb{N}}$ of $G\wr S_{\bullet}$, w…

Let $m\in\mathbb{N}$ and let … For $\gamma\in\mathcal{F}$, define … and let $\operatorname{Crit}(\gamma)$ be the subgroups achieving this maximum. Puder–Zimhony's conjecture. If…

Let $\Sigma_n\curvearrowright X_n$ be an asymptotically oligomorphic sequence of group actions, let $G$ be a compact group, and let $\phi\in\widehat G\setminus\\{\mathbf{1}\\}$. Fo…

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…

Let $K$ be a finite field with $|K|=q$, let $\mathbb{F}$ be a free group, and let $w\in\mathbb{F}$. Let $\mathcal{R}$ be the algebra of character-like functions on the groups…

Let $K$ be a finite field with $|K|=q$, let $\mathbb{F}$ be a free group, and let $w\in\mathbb{F}$. Write $\pi=\pi_q(w)$ for the $q$-primitivity rank of $w$, and let…

Let $G=\{G(N)\}_N$ be a natural family of groups, and let $\chi=\{\chi_N\}_{N\geq N_0}$ be a stable irreducible character with $\chi_N\in\widehat{G(N)}$. Let $\pi(w)$ denote the pr…

Let $\mathbf{F}$ be the free group, and let $w_{1},w_{2}\in\mathbf{F}$. For a compact group $G$, write $\mu_{w}$ for the probability measure on $G$ obtained by evaluating $w$ on a…

Let $\mathbf{F}_r$ be the free group of rank $r$, and let $w\in\mathbf{F}_r$ induce a probability measure on each compact group via its word map. Word-measure rigidity conjecture.…

Let $\mathbf{F}_{r}$ be the free group of rank $r$, and let $w_{1},w_{2}\in\mathbf{F}_{r}$ be words. For a compact group, evaluate each word on independent Haar-random elements to…