Strong conciseness of coprime commutators in profinite groups

Abstract: Let $G$ be a profinite group. The coprime commutators $\gamma_j^*$ and $\delta_j^*$ are defined as follows. Every element of $G$ is both a $\gamma_1^*$-value and a $\delta_0^*$-value. For $j\geq 2$, let $X$ be the set of all elements of $G$ that are powers of $\gamma_{j-1}^*$-values. An element $a$ is a $\gamma_j^*$-value if there exist $x\in X$ and $g\in G$ such that $a=[x,g]$ and $(|x|,|g|)=1$. For $j\geq 1$, let $Y$ be the set of all elements of $G$ that are powers of $\delta_{j-1}^*$-values. The element $a$ is a $\delta_j^*$-value if there exist $x,y\in Y$ such that $a=[x,y]$ and $(|x|,|y|)=1$. In this paper we establish the following results. A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$ such that the set of $\gamma_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$. A profinite group $G$ is finite-by-(prosoluble of Fitting height at most $k$) if and only if there is $k$ such that the set of $\delta_k^*$-values in $G$ has cardinality less than $2^{\aleph_0}$.