$\mathrm{\varkappa }$-Fréchet--Urysohn and Baire subgroups of free topological groups

A topological space is $\mathrm{\varkappa }$-Fréchet–Urysohn if every point in the closure of an open set $U$ is the limit of a sequence of points of $U$. We prove that every $\mathrm{\varkappa }$-Fréchet–Urysohn subgroup and every Baire subgroup of the free topological group on a Tychonoff space $X$ in any full variety of topological groups (in particular, of the free topological group $F(X)$, the free Abelian topological group $A(X)$, and the free Boolean topological group $B(X)$) is discrete. In particular, each of these groups is $\mathrm{\varkappa }$-Fréchet–Urysohn (or Baire) only if $X$ is discrete, and all their metrizable and even Fréchet–Urysohn subgroups are discrete. Along the way we characterize strongly $\mathrm{\varkappa }$-Fréchet–Urysohn spaces in the spirit of Michael’s theorem and study countable families of closed sets absorbing convergent sequences in $\mathrm{\varkappa }$-Fréchet–Urysohn spaces and groups.