4 problems

Let $l\in\mathbb{N}$. A non-singular mixed identity for a group $G$ is a non-trivial word in the free product $G\ast\mathbf{F}_r$ that is satisfied under the relevant substitutions…

For a positive integer $d$, let $\operatorname{F}(X_1,\ldots,X_d)$ be the free group on $X_1,\ldots,X_d$, and let $w\in\operatorname{F}(X_1,\ldots,X_d)$ be a nontrivial word. For a…

Let $D$ be a division ring with center $F$, and let $\operatorname{GL}_n(D)$ be the general linear group of degree $n$ over $D$. If $n\geq 2$, assume that $D$ is not a locally fini…

Let $w$ be a group-word and $G$ a profinite group. Suppose that the set of $w$-values in $G$ is countable. Identity conjecture. Then $G$ satisfies a nontrivial group identity. The…